Asynchronous logic with intermediate value between data and null values

ABSTRACT

A switching element and method for asynchronous logic switches an output signal according to a switching-logic relationship between or among input signals. Input signals may assume at least a DATA value, a NULL value, and an INTERMEDIATE value. Input signals may also assume multiple DATA values. The element and method generates an output which assumes a NULL value when all input signals are NULL, and assumes DATA and INTERMEDIATE values in accordance with transform rules. Output signals may also assume an INTERMEDIATE value. DATA, NULL and INTERMEDIATE values may be encoded on to a number of signal lines. Transform rules may be threshold switching rules, where the output switches to a DATA output when a number of DATA inputs is greater than a threshold.

This is a continuation of application Ser. No. 08/220,636, filed Mar.31, 1994 now U.S. Pat. No. 5,664,212; which is a continuation ofapplication Ser. No. 08,074,288, filed Jun. 8, 1993, issued Apr. 19,1994, as U.S. Pat. No. 5,305,463; which is a continuation of applicationSer. No. 07/702,016, filed May 17, 1991, abandoned.

SPECIFICATION BACKGROUND OF THE INVENTION

This invention relates to information processing systems and methods formanipulating and resolving data, and particularly to computer systemsand methods. The invention is particularly useful for building circuits,and for building processors utilizing a plurality of such circuits.

Traditional electronic logic circuits are composed of continuouslyacting logic elements which are continuously asserting a potentiallylegitimate result. When new input data is presented to a circuit, theresult asserted by the circuit might change several times before settingto the correct result. In general it is not possible to determine, interms of the circuit output itself, when the correct result is assertedby the circuit.

The determination of completion has generally been accomplished by arepresentation external to the circuit, most often a system clock. Othertypes of external representations have also been used, such as a delayline associated with each circuit. Such external systems are provided toindicate when the output of the circuit is VALID. They do so by allowingsufficient time for the logic circuit to settle to a correct resultbefore declaring the result values VALID.

External timing of the logic circuit requires that the control and logicrepresentations be carefully engineered to coordinate their timingcharacteristics. Because they must be coordinated or synchronized, suchsystems are typically referred to as synchronous systems. In the case ofa system clock, the circuit has to be carefully designed so that it isassured of settling to a correct result within one clock period.Similarly, a delay line must be long enough to accommodate the timing ofthe circuit and it must also be guaranteed that its delayed controlsignal will be stable for a long enough time. Such synchronizationconsiderations place significant complications on system design.

The existing technology of speed independent or Muller circuits does notpostulate a NULL value integrated into the primitive transform elements.It relies instead on Boolean logic gates carefully arranged to providethe whole circuit with a specific resolution behavior. This cannot,however, eliminate all possible hazards due to circuit element delays.Transmission elements can introduce delays that could cause incorrectfunction of the circuit. The existing technology is not logicallycomplete in that physical timing characteristics of the circuit elementstill have to be considered in any circuit design.

Speed independent circuits put the burden of completion integrity on theconfiguration of the circuit itself and cannot achieve a purely logicalexpression of the circuit. The NULL convention puts the burden ofcompletion integrity on the primitive transform elements. This allows apurely logical expression of a circuit quite independent of the physicaltiming behavior of all circuit elements.

A known concept using a non-data representation as a control means isthe spacer code of dual rail encoding associated with speed independentcircuits. Dual rail encoding, however, is an interface signalingprotocol between circuits and is not a concept associated with theinternal organization of the circuits themselves.

Despite the need for a system or environment in the art which enablesautonomously acting and coordinated logic circuits to implementindependently acting and locally controlled process representationswithout the need for external control representations, and whichovercomes the limitations and problems of the prior art, none insofar asis known has been proposed or developed.

Accordingly, it is an object of the present invention to provide aninformation processing system and method for manipulating and resolvingdata. Another object of the invention is to provide a system and methodwhich is useful for constructing information processing units andmembers such as circuits and gates and for constructing configurableprocessors utilizing a plurality of such circuits and gates.

Yet another object of this invention is to provide a system and methodwhich enables processors having autonomously acting and coordinatedlogic circuits to implement independently acting locally controlledprocess representations without the need for external controlrepresentations. A further object of the invention is to provide asystem and method as described above which utilizes the representationof control as a value with respect to the logic circuits and gatesthemselves.

Still another object of the invention is to provide a system and methodof utilizing a null convention in logic and processor design andfunction.

SUMMARY OF THE INVENTION

The present invention provides an information processing system formanipulating and resolving data, which has at least one informationprocessing unit for resolving combinations of data and non-data values,for example a logic circuit. The information processing unit or unitseach have at least one information processing member, for example alogic gate, also for resolving data. The unit further has a plurality ofinformation transmission elements, for example conductors, each elementtransmitting the data to and from the one or more members or units.

The information transmission elements may be electrical, optical or anyother transmission means known in the art. Data manipulated and resolvedby the system and the system components described above consist ofvalues which, for example, may represent physical states on or in theelements, members and units. Such physical states represent voltage,optical energy or any other medium which may be utilized to conveyinformation pertaining to velocity, temperature, or angular position forexample. Importantly, the system and its components also transmit andresolve non-data values.

Each information processing member and unit has one or more informationtransmission elements connected to itself for both input or presentationof values, and for output or assertion of result values. And, althoughelements are capable of transmitting only one values at a time, membersand units are capable of resolving combinations of values (information)which are presented either individually over time via a single element,or simultaneously via multiple elements.

Allowed values are those which are resolvable by information processingmembers and units, and consist of at least one data value and at leastone non-data value, at least one non-data value being a null value. Theset of data values includes a single value or two values, for example,such as is used in traditional binary logic. In addition to the nullvalue, one or more intermediate values, which are distinct from the nullvalue and the data values, may be included in the set or group ofallowed values.

Accordingly, in one embodiment of the information processing system ofthe present invention there are two allowed values, the first allowedvalues being a data value and the second allowed value being a nullvalue. In another embodiment, there are three allowed values, the firstallowed value being a data value, the second allowed value being a datavalue, and the third allowed value being a null value. In still anotherembodiment, there are three allowed values, the first allowed valuebeing a data value, the second allowed value being an intermediatevalue, and the third allowed value being a null value. In a finalembodiment of the present invention, there are four allowed values, thefirst allowed value being a data value, the second allowed value being adata value, the fourth allowed value being an intermediate value, andthe fourth allowed value being a null value.

Each information processing unit maps from combinations of valuespresented to it to combinations of values to be asserted by it. Toachieve play-through, the information processing members resolve valuesby asserting a value for each combination of values presented to it,such that (1) for valid combinations of presented values the assertedvalue is a data value dependent upon the particular combination ofpresented values, and (2) for invalid combinations of presented valuesthe asserted value is a null value.

To achieve hysteresis, the information processing members resolve valuesby asserting a value for each combination of values presented such that(1) for valid combinations of presented values the asserted value is adata value dependent upon the particular combination of presented valueswhich remains asserted until the combination of presented values becomesall-null, and (2) for all-null combinations of presented values theasserted value is a null value which remains asserted until thecombination of presented values becomes valid.

With respect to intermediate value resolution, information processingmembers resolve values by asserting a value for each combination ofvalues presented such that (1) for valid combinations of presentedvalues the asserted value is a data value dependent on the particularcombination of presented values, (2) for combinations of valuesincluding intermediate values the asserted value is an intermediatevalue and (3) for all-null combinations of presented values the assertedvalue is a null value.

The information processing unit cycles through resolution andnon-resolution states to allow determination of the unit's (1)completion of a data resolution and (2) readiness to perform anotherdata resolution. A resolution state occurs when the unit is presentedwith a valid combination of values and is asserting a valid combinationof values. A non-resolution stat occurs when the unit is presented withan all-null combination of values and is asserting an all-nullcombination of values.

The information processing unit can perform a specific data resolutionfunction by asserting a predetermined combination of values for eachcombination of presented values. The unit can further store values byasserting a combination of values equal to a previously presentedcombination of values. The unit can select values by asserting acombination of values which are a subset of a first combination ofpresented values relative to a second combination of presented values,and can also distribute data by asserting a combination of values equalto a first combination of presented values as a subset of it'scombination of asserted values relative to a second combination ofpresented values.

The information processing system further comprises bounding means forasynchronously coordinating value presentation to the informationprocessing units. Each bounding means comprises a null-valid detectorfor determining a unit's (1) completion of a data resolution and (2)readiness to perform another data resolution; means for storing valuecombinations, and means for communicating with other bounding means.

The null-valid detector is preferably an information processing unitwhich asserts a null value when it's combination of presented values isall-null and continues asserting a null value until it's combination ofpresented values becomes valid, whereupon it asserts a data value andcontinues asserting a data value until it's combination of assertedvalues becomes all-null. The means for storing is preferably a unitwhich asserts a combination of values equal to a previously presentedcombination of values. The communication means comprises:

(a) means for informing all existing preceding bounding means that afirst null-valid detector has detected a combination of presentedvalues, the valid combination of presented values has been stored in thestoring means, and that all existing preceding bounding means can nowassert an all-null combination of values;

(b) means for informing all existing preceding bounding means that thefirst null-valid detector has detected an all-null combination ofpresented values and all existing preceding bounding means can nowassert a valid combination of values;

(c) means for detecting that all existing succeeding bounding means havedetected and stored a valid combination of values resulting from thevalid combination of values stored in the storing means and asserted bythe bounding means, whereupon an all-null combination of values can beasserted by the bounding means; and

(d) means for detecting that all existing succeeding bounding means havedetected an all-null combination of values asserted by the boundingmeans, whereupon a valid combination of values can be asserted by thebounding means.

A generally configurable information processing system is constructed ofan information processing unit for resolving combinations of values, aunit for storing combinations of values, a unit for configuring valuepresentation relationships relative to a second combination of values,and bounding means for asynchronously coordinating value presentationbetween units.

The configuring units, for example a distributor or selector, configurevalue presentation relationships among at least one resolving unit andat least one storing unit relative to a combination of directive values,for example program instructions, asserted by a first bounding means andpresented as the second combination of values of the configuring unit.

A resolution configuration exists during the presentation of a validcombination of directive values to the configuring unit resulting in thepresentation of a valid combination of values to the storing unit. Anon-resolution configuration exists during the presentation of anall-null combination of directive values to the configuring unitresulting in the presentation of an all-null combination of values tothe storing unit.

Data resolution is accomplished by a progression of alternatingresolution configurations and non-resolution configurations relative toa progression of combinations of directive values.

These and other benefits of this invention will become clear from thefollowing description by reference to the drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the general structure of null convention logic circuitsbounded by boundary elements,

FIG. 2 shows a network of association relationships with no cycles,

FIG. 3 illustrates a variable association rule as a bag that isolatesvariables and also keeps them together,

FIG. 4 illustrates the precedence of occurrence relationships amonginteractions for the example process,

FIG. 5 shows a rigid structure of interacting variables that might beexpected to express the example process,

FIG. 6 illustrates the relationships between input variables and theresult asserting variable,

FIG. 7 illustrates the difficulty directionality of interaction betweendirectly associated variables,

FIG. 8 illustrates how many isolation variables must be between any twovariables that assert the same value set,

FIG. 9 illustrates how variables can be arranged in a standardassociation unit called an interaction locus that isolates identicalinput and result value sets,

FIG. 10 shows one way to graphically represent an interaction locus,

FIG. 11 shows how several interaction loci can be directly associated torender a larger process expression,

FIG. 12 shows that the standard graphic form of representing logiccircuits is identical to the more general form developed from thetheory,

FIG. 13 is a directly associated expression with interaction loci whichstill cannot determine its own completion,

FIG. 14 is a configuration of interaction loci that will work with aNULL value added to the value sets of the interaction loci,

FIG. 15 illustrates a configuration of rotation loci to convert a giveninput name to a standard recognition name,

FIG. 16 is the input section of an expression that will recognize all ofthe possible input names of the process,

FIG. 17 is the complete expression that will transform each possible 4variable-4 value input name to a specific 4 variable-4 value resultname,

FIG. 18 shows how the input name recognition stage can be optimized,

FIG. 19 shows how the result collection and assertion stage can beoptimized,

FIG. 20 shows the complete optimized expression to recognize 4variable-4 value input names and assert 4 variable-4 value result names,

FIGS. 21a-c illustrate how the expression of the example process can beexpressed if custom interaction loci are allowed,

FIGS. 22a-b illustrate how the expression can be further optimized byreassigning the coding of the input names,

FIGS. 23a-b illustrate the expression of the example process if 6 valuesare available.

FIGS. 24a-b illustrate the expression of the example process if 9 valuesare available,

FIGS. 25a-b show that the expression becomes a pure variable expressionit 15 values are available,

FIGS. 26a-b show the transform table for the example expression if only2 values are available,

FIG. 27 shows the example process as a, prior art, standard logiccircuit,

FIGS. 28a-b show the 2 value NULL convention expression of the simplerexample process of FIG. 12,

FIG. 29 shows the basic example process expressed as 2 value NULLconvention logic,

FIG. 30 illustrates the general form for a generally configurableprocess,

FIGS. 31a and 31 b show a variable structure and transform rule set toimplement a memory element,

FIGS. 32a-b show a variable structure and transform rule set toimplement a selector element

FIG. 33 illustrates the composite name form for a generally configurableprocessor,

FIG. 34 illustrates a selector element structure,

FIGS. 35a-b show a variable structure and transform rule set toimplement a distributor element,

FIG. 36 illustrates a distributor element structure,

FIG. 37 illustrates the portion of a generally configurable processexpression that configures association relationships for eachinteraction,

FIG. 38 shows the structure of the boundary element,

FIGS. 39a-b show a variable structure and transform rule set toimplement a NULL-VALID detector element,

FIG. 40 shows how a NULL-VALID detector can be cascaded to accommodateany size name,

FIG. 41 shows how boundary elements communicate to synchronize valueflow,

FIG. 42 is the transform table for the protocol section of the boundaryelement,

FIG. 43 shows the value change timing relationships among the variablesof the boundary element,

FIG. 44 illustrates how name resolving expressions are inserted betweenboundary elements to form larger expressions,

FIG. 45 illustrates how boundary elements can be pipelined,

FIG. 46 shows a fan in configuration of boundary elements,

FIG. 47 shows a fan out configuration of boundary elements,

FIG. 48 shows the complete generally configurable process expression,

FIG. 49 is an example process expression to mapped into the generallyconfigurable process,

FIG. 50 shows the sequence thread of interactions for the generallyconfigurable process to perform one at a time,

FIG. 51 shows an expression bounded by a boundary element that willdetermine its resolution completion,

FIGS. 52a-b show a typical variable structure and transform rule setthat does not guarantee the integrity of the NULL value wavefront,

FIG. 53 shows an expression for which the NULL value wavefront is notcompletely determinable,

FIGS. 54a-b show a variable structure and transform rule set thatprovides a hysteresis solution to the NULL value wavefront problem,

FIG. 55 shows the feedback configuration to provide a 1 DATA valuethreshold locus with hysteresis to make the NULL wavefront determinable,

FIG. 56 illustrates an expression with encoded values and hysteresisinteraction loci,

FIG. 57 shows how a NULL-VALID detector can be specified withINTERMEDIATE values,

FIG. 58 shows a threshold 3 INTERMEDIATE value interaction locus with 3input variables,

FIG. 59 shows a threshold 2 INTERMEDIATE value interaction locus with 3input variables,

FIG. 60 illustrates an intermediate expression with 2 composite inputvariables,

FIG. 61 illustrates a threshold 2 expression with 3 composite inputvariables, and

FIG. 62 shows a threshold 3 expression with 3 composite input variables.

DESCRIPTION OF THE PREFERRED EMBODIMENT I. Overview

The present invention provides a new information processing system orrepresentational environment which eliminates the need forsynchronization means such as clocks, delay lines, or peripheraltransition signaling means. The system itself provides a NULL (N) valuewhich represents control. The NULL value is utilized as a flag ofinvalidity such that when it is asserted, data on a particular input isnot valid. If, on the other hand, a nonNULL or data value is asserted,then the input is valid. The NULL value is added to the traditionalbinary values of TRUE (1) and FALSE (0) to yield a three-value logicsystem. Alternatively, the NULL value can be added to a single datavalue to yield a two-value logic system. In this case a dual-railencoding can be used to obtain two data values necessary forconventional logic representations.

Since the representation of control is added as an extra value to thelogic of an information processing circuit or unit itself, instead of inthe form of an independent external representation, the circuit is madeautonomously determinable and the engineering coordination andsynchronization complications associated with two independentrepresentations are thus avoided. The circuits and informationprocessing gates or members provided thereby autonomously express theirown completion of data resolution, thus enabling them to operatecompletely asynchronously in relation to each other.

An essential characteristic of such asynchronous transmission control isthat events are controlled locally and are no longer constrained tooccur at specific times or during discrete intervals. Accordingly,circuit design is made more simple and straightforward. Moreover, thiscan be accomplished using standard binary logic.

As shown for example in the table below, the addition of a NULL value toan existing logic circuit does not alter the established logic of therepresentation. For example, standard two-value logic gates are replacedby corresponding three-value logic gates. The NULL value indicatesmeaninglessness within the representation.

The NULL value is added to the existing truth tables so that if eitherinput to a gate is NULL, the output will be NULL. An input to a gate isVALID only when both inputs are nonNULL. An INVALID input is one with atleast one NULL value. As soon as both inputs are nonNULL, the input isvalid and a nonNULL (DATA) value will be asserted. No gate will assert aVALID output unless it's input is VALID. Each gate exhibits a distinctresolution event (the output transition from NULL to nonNULL) whenever aVALID input is presented to it. Each gate indicates the completion ofit's own logical transformation process.

A practical example of the use of the NULL value convention is an ANDlogic gate implemented with a transistor circuit and having the threevoltage values of +5, 0, and −5. If 0 is assigned to represent the NULLvalue, the −5 and +5 are VALID data values. When 0 is asserted to eitherinput of the AND gate, the output will be forced to 0. If non-0 values(−5 or +5) are asserted on both inputs to the gate, the output is thelogical AND of the two values the non-0 voltages represent. If 0 isnever asserted (except perhaps in transition), then the gate acts justlike a standard logic gate.

As previously discussed, the NULL value convention can also supportsingle-data-value representations. Referring to the table below, thisform of representation only has two values, NULL (N) and DATA (D).Two-value NULL convention logic is related to threshold logic. Sincethere is only one nonNULL value, data must be represented by quantitiesof DATA values. In the truth tables, D is DATA and N is NULL. For thethreshold 1 table a single DATA value will set the output to DATA. Forthe threshold 2 table both inputs must be DATA for the output to beDATA.

In addition to utilizing the NULL value at the gate level of a circuit,the NULL value may also be utilized at the circuit level to indicate theproper transfer of data between circuits. The representation elementthat manages the presentation of data between two NULL convention logiccircuits is called a boundary element. As shown in FIG. 1 nullconvention logic circuits bounded by such elements A and B behave asindependently proceedable units that accept data, process it, and thenpass it on to other circuits or units. The input data, in addition toexpressing values, is either VALID or all-NULL, thus incorporating thecontrol information necessary for determining it's representationalvalidity

The assertion and presentation of value combinations involves ahandshake protocol convention between two circuits. In the protocol oftwo control variables each with two values, the input data itself servesas one of the control variables. The input data in addition toexpressing the values of the data is either VALID or all-NULL. These twostates of VALID and all-NULL in the data itself provide two values forone of the handshake variables. So the data stream itself becomes one ofthe handshake variables.

An input is received and stored in a current boundary element and stablyasserted until the next boundary element can receive it. Then thecurrent boundary element is freed up to receive another input. Dataflows through a representation as packets from boundary element toboundary element. As discussed with respect to FIG. 1, between boundaryelements there can be any NULL convention logic circuit that willperform some transform on the data. A NULL convention logic circuitbounded by boundary elements will cycle through completely NULL statesand completely VALID states so that the NULL convention criteria thatallows determination of the completion of the resolution of a NULLconvention logic circuit is satisfied.

Boundary elements can be associated in various structures, for example asimple pipeline structure. The pipeline is completely autonomous. AVALID input presented to the first element in the pipeline will begin asequence of interactions that will propagate the data from element toelement all the way through the pipeline. As each element sees a VALIDinput, it will receive it and assert it to the next element in thepipeline. Several data packets can be simultaneously propagating throughthe NULL convention pipeline just like any other pipeline. Thepropagation rate of the pipeline is determined by the longesttransmission delay between two boundary elements. Boundary elements canalso be associated in fan-in, fan-out, and circular configurations.

A VALID input at a circuit representing a desired process utilizingassociated boundary elements will trigger the progression of events thatis the represented process. As the events proceed, the circuit resetsitself to be triggered again. As completed processing results arrive atthe output of the circuit, their completion is signaled by the assertionat the output of the values comprising the completed processing resultsthemselves.

Boundary elements are preferably used to partition a representation intodiscrete independently proceedable units that may be complex internally,but that have relatively simple interfaces between them. This allowsmany representational elements to be simultaneously operating,increasing the throughput of the circuit.

The system of the present invention can process data with NULL valuesvia its information processing members (logic gates) that resolvecombinations of data values. In contrast, the prior art technology canmerely transmit an indeterminant value along with data values overtransmission elements (wires, for example). As a result of thisadvantage the system is speed independent at every level in that theinformation processing units (circuits) and members (gates) of thissystem report their own completion.

The NULL convention logic system provides a representational environmentin which autonomously acting and coordinated NULL convention logiccircuits can implement independently acting and locally controlledprocess representations. External global control representations are notneeded. The system of this invention is applicable to digital computers,telephone switching systems, and in a variety of control systems,particularly those amenable to asynchronous processing and control.

In the description below, process expression is discussed first ingeneral terms (11), and then in greater detail to demonstrate the needfor the NULL value convention in process expression, in terms of aprocess for resolving data (111). The generally configurable processexpression is then introduced, and particularly with respect to a systemfor manipulating and resolving data (IV). Finally the completionintegrity of successive resolution cycles and the intermediate non-datavalue is discussed (V).

II. A Model of Process Expression

This discussion introduces a model of process expression that focuses onthe nature of process itself. The model will lay a foundation forexploring process expression.

Process is defined in very general terms then the invocation model ofprocess expression is introduced. A series of examples present the twodomains of expression within the model and illustrate the fundamentalprinciples of process expression.

A. Process

Process is change occurring to existential entities due to interactionsamong those entities. Process begins with the possibility ofinteractions among entities. There may be many possible interactions buteach process is the actualization of only one possibility. Process is aconfluence of possibility and actuality. All of the details of eachpossible interaction may be prespecified in detail before the fact, butwhich possible interaction will actually occur cannot be specified untilthe time of occurrence. The possibility and the actuality of a processare inherently separate aspects of the process and must be separatelyexpressed. These aspects will be called the possibility expression andthe actuality expression.

The possibility expression is an inherently incomplete specification ofa process. It must defer as an open question the specification of whichof the possible interactions will occur. The actuality expressionprovides the answer to this question and by combining with a possibilityexpression completes the specification of a particular interaction. Thepossibility and actuality expressions must correspond in the sense thatthe answer of the actuality expression must fulfill the question of thepossibility expression. A computer and program is the possibilityexpression and data is the actuality expression.

B. The Invocation Model

There must be existential entities. The entities must suppport aproperty of changeability in addition to their own existence. There mustbe a means of specifying how these entities associate to interact andthere must be a means of specifying what changes accrue frominteraction.

The existential entities will be called variables. A value is thenonexistential property of a variable that can change duringinteraction. Every variable is always asserting exactly one value.Variable association rules specify which variables can associate witheach other to interact. Value transform rules specify which values caninteract and how these values will change. The model consists ofvariables, values, variable association rules and value transform rules.Variables interactively associate according to variable associationrules. Values asserted by the variables change according to valuetransform rules.

1. Variable Association Rules

The variable association rules specify which variables can interact. Iftwo or more variables are associated then those variables areinteractively proximate. If variables are not associated by anyassociation rule then they are not proximate and cannot interact.

The association rules differentiate the variables by specifying exactlyhow each variable is interactively proximate with one or more othervariables. Each variable has a place in the process universe and each isdifferentiated by its place in relation to the other variables. If anassociation rule states that all variables are or may be in interactiveproximity then each variable is in no particular place in relation tothe other variables and there is no way to differentiate one variablefrom another. If an association rule states that no variables areassociated then it doesn't matter whether variables can bedifferentiated, they cannot interact and are effectively nonexistent.

2. Value Transform Rules

The value transform rules specify what changes occur to interactingvalues. A value transform rule specifies a combination of values as thename of the rule and a set of result values. The combination of valuesthat forms the name of a value transform rule specifies that thatcombination of values is interactively proximate and will interact. If acombination of values does not form the name of any value transform rulethen that combination of values is not proximate and they will notinteract. The result values of the rule specify the values that willresult from the interaction.

Similarly to variables, values receive their differentiation and placein the interaction universe from the value transform rules. Only valuesthat appear in value transform rule names are interactively proximateand then only in the combinations specified by the names. Valuesappearing both in value transform rule names and also as result valuesin other value transform rules form relationships among value transformrules that forge the structure of the value interaction universe andestablish each value's place in it. A value is differentiated by itsplace in relation to all the other values in the process universe asspecified by the set of value transform rules for the process.

3. Primitive Process Interaction

A primitive process interaction occurs when an actuality expressionarises and an interaction rule name is formed. This occurs when at leasttwo variables are in interactive proximity and their values are ininteractive proximity (the values form the name of a value transformrule). If two variables are in interactive proximity but their valuesare not in interactive proximity (they do not form a value transformrule name) no interaction will occur. Conversely values can beinteractively proximate but if their asserting variables are notinteractively proximate no interaction will occur.

The possibility expression of the process is a set of value transformrules and the existence of the variables. Each value transform rule isone possible interaction. The question of which possible interactionwill proceed is posed by the names of the rules. The question is whichrule name will be formed.

The answer is provided when two or more variables become interactivelyproximate and their values form the name of one of the value transformrules. Like the sorcerer invoking his demons by name to perform hismagical transformations the named rule is invoked and the values aretransformed. Hence the invocation model. The values of interactivelyproximate variables forming a transform rule name is the actualityexpression for the interaction. It is the delivery of the deferredspecification, the answer to the question of which of the possibleinteractions will proceed.

4. Interaction Composition

The next stage of process expression beyond a single interaction is aprogression of dependent interactions. The dependency relationshipsamong the interactions are expressed as name formation dependencyrelationships. The name formed for one interaction depends on the resultvalues of one or more other interactions. The dependency relationshipcan be expressed as direct association relationship between each valueof a name of one interaction and a result from another interaction thatprovides that value. It is still not possible to predetermine whichtransform name will be formed for each interaction but it can bepredetermined where the interaction's transform name will come fromwithin the expression.

These dependency relationships form a network of associationrelationships of result values to name values among the interactions.This network of association relationships provides each interaction witha place in relation to all the other interactions.

FIG. 2 shows a network of association relationships with no cycles.Although any configuration of association relationships is possible,only association networks with no cycles are discussed in detail. Otherassociation configurations will become apparent to those skilled in theart.

Each individual interaction is still expressed by its own possibilityexpression consisting of a set of value transform rules. Theseindividual possibility expressions are combined into a larger compositepossibility expression through the name formation dependencyrelationships. Most of the interactions receive their transform namevalues from specified local places in the structure of associations. Afew name values, however, are not associated to any local places andthey must come from someplace external to the structure of associationrelationships. These unassociated name values taken collectively are thecomposite actuality expression for the composite possibility expression.A composite actuality expression will also be referred to as the inputname for the composite possibility expression. The result values of thecomposite possibility expression will be referred to as the result name.

The composite possibility expression is a complete stand alonespecification of possibilities dependent only on its composite actualityexpression. It is a larger self contained unit of expression (apossibility expression and an actuality expression) than theinteraction.

These name formation dependency association relationships amonginteractions can be expressed within the invocation model either interms of value transform rules or in terms of variable associationrules. This means that the model possesses two quite different domainsof expression.

C. Two Domains of Expression

The value transform rules and the variable association rules provide themodel with two domains of differentiation and interactive association. Aprocess can be expressed almost exclusively in terms of differentiationand interactive association of values by value transform rules or almostexclusively in terms of differentiation and interactive association ofvariables by variable association rules. Between these two extremesthere is a large intermediate landscape of cooperative expression withgradations of expression from each domain.

1. Pure Value Expression

At one extreme all differentiation and interactive association is interms of values and specified by value transform rules. There is nodifferentiation in terms of variable association rules. Variables areall simultaneously associated or they associate indiscriminately. Thereis no way from the point of view of interaction to tell one variablefrom another. All differentiation in the expression is by differentvalues. All interactive association is specified by correspondence ofunique values. This form of expression will be called a pure valueexpression.

A natural example of this form of expression is a chemical reaction. Themolecules are variables. The composition of each molecule determines itsinteraction possibilities and is its value. A liquid or gas is nature'sversion of the association rule that specifies that there is nodifferentiation of variables. The variables (molecules) willindiscriminately associate in the liquid or gas and there is no way totell one variable from another except by the value it asserts. Themolecules get together forming transform rule names and interact to formnew molecular values. Another example of this form of expression issubatomic particles. The variables (particles) indiscriminatelyassociate in free space and there is no way to tell one variable(particle) from another except by the value it asserts. The particlesget together forming transform rule names and interact to form newparticles and composite structures.

2. Pure Variable Expression

At the other extreme all differentiation and interactive association isin terms of variables and specified by variable association rules. Thereis no differentiation in terms of value transform rules. There is a setof values and a set of value transform rules for all possible inputnames that those values can form. This means that all values areconstantly in interactive proximity. There is no way from the point ofview of interaction to tell one value from another. All differentiationin the expression is by variable association. All interactiveassociation is specified by specifically associated variables. This formof expression will be called a pure variable expression.

A natural example of this form of expression is a network of neurons.The artificial computer is also primarily of this form.

Within the model there must always be a bit of each domain in everyprocess expression. There must always be variables that associate andthere must always be values that transform. Nevertheless the two ends ofthe territory will be referred to as pure value and as pure variable.

3. The Intermediate Expression Territory

Expression in the intermediate territory between pure value and purevariable is a cooperative endeavor between variable differentiation andvalue differentiation. A very simple example of this cooperativeexpression is the representation of numbers with different bases. Basetwo numbers have only two values and very simple value transform rules(logical truth tables) but have lots of digits that must be properlyassociated. Base ten numbers have ten values and larger, more complexvalue transform rules (traditional addition and multiplication tables)but have fewer digits to be associated. These gradations of expressionranging from mostly in terms of value differentiation and valuetransform rules to mostly in terms of variable differentiation andvariable association rules will for convenience be called simply valueexpressions and variable expressions.

D. Value Expression

In a value expression there is no differentiation of variables. Alldifferentiation is in terms of values and specified by value transformrules. Transform rules will be represented in the format of thetransform name followed by the result values enclosed in brackets, forexample:

“name [result, result, . . . ].”

Consider an example expression with 4 differentiated values A, B, C andD and two value transform rules that interactively associate thosevalues. A and B interact and transform into D. C and D interact andtransform into B. The two transform rules for the example are:

AB[D] or BA[D]

CD[B] or DC[B]

The linear string representation makes it appear that ordering issignificant when actually it is not. The input names AB and BA are equaland are both specified in this example. There can be no orderrelationships among the values because there can be no orderrelationships among the variables when they are interactively proximate.They are simply two values interacting. AB and BA are the same inputname.

The value transform rule is the simplest most primitive form of processexpression. No part of the rule can be reduced to simpler terms or bemore precisely stated. The values are primitive and indivisible and theinteraction result is complete in itself and unambiguous.

The value transform rules specify completely who does what with whom. Asfar as the variables are concerned anybody can do anything with anybodybut in terms of the values only As can interact with Bs and only Cs caninteract with Ds. As cannot interact with Ds nor Bs with Cs and soforth. There cannot be the A of this variable and the A of that variablebecause there is no this variable or that variable. Value difference isthe only way to tell anything apart for this expression. A valueexpression can be elaborated and extended to more expressionalcomplexity by adding more values and more value transform rules.

A pure value expression differentiates all parts of the expression byvalues. The input values form the name of a transform rule. The resultspecified by the transform rule is a different value from the inputvalues so there is no ambiguity about when the interaction is complete.The input values disappear and the result values appear. The new valuecan form a new unique input name with other values and a new interactioncan occur which results in new values which can form further uniqueinput names and so on. The process progresses in a succession offulfilled interaction possibilities determined by the formability ofunique transform rule names. Each interaction possibility is dependenton one or more previous interactions to provide the values to form itsinput name. The set of transform rules below illustrates a valueexpression that proceeds in stages of unique input name formationdependencies.

AB[C, D]

CC[D, E]

DD[G, F]

EF[F, G]

GG[X, Y]

If there are some As and Bs present they will begin interacting andgenerating Cs and Ds. The CC and DD interaction cannot occur until thereare some Cs and Ds available. The CC and DD interactions will generateEs, Fs and Gs which will form the input names of the EF and GG,transforms will interact and finally produce Xs and Ys. For this processthe possibility expression is the set of value transform rules and thevariable association rule that all variables are or will becomeinteractively proximate. The actuality expression is the presence of theAs and Bs.

The process is resolved in distinct discrete stages. The completion ofeach stage is established by the presence of result values and theabsence of input values for that stage. When Xs and Ys appear theresolution of the process is completed. It is deterministic anddirectional. It can't run backwards and start generating As and Bs. Allof these characteristics are determined solely by the differentiationand specific association of values as specified by the value transformrules.

1. Traditional Computation with Value Expressions

The value expression form is unfamiliar territory in the contemplationof process expression and computation. Traditional computation expressesresults (numbers) in the same form as the input was expressed (numbers).For the value expression this means that the results are expressed withthe same values. This can lead to ambiguities in the direction ofprogression of resolution and indeterminability of completion. But thevalue expression form can accommodate these issues. The followingexamples will illustrate that the while the pure value end of theprocess expression landscape is associated mainly with naturalexpressions such as chemistry, physics and biology it is a fully capablecomputation environment.

There is in fact an important value expression system in the history ofcomputational thought. This is the Roman numeral system without thesubtractive principle. (This means that 9 is VIIII instead of IX). Thedigits of Roman numerals are generally presented in a certain order forconvenient reading but without the subtractive principle the order hasno significance to their meaning. No matter what arrangement the digitsare presented in they represent the same number. Each digit of a numberis a variable, each variable has a value. The variables have noparticular association relationships among themselves. The meaning ofeach digit is differentiated entirely by its value. The magnitude of thenumber is determined solely by the values present.

Possible values are: M,D,C,L,X,V,I

Transform rules are:

IIIII [V]

VV [X]

XXXXX [L]

LL [C]

CCCCC [D]

DD [M]

The only interactions possible in this Roman numeral expression arethose specified in the value transform rules. Vs can't interact with Xsbecause there is no rule with the appropriate name. The above valuetransform rules are a complete expression of the process for addition inRoman numerals. Given two Roman numbers these rules will reduce them toa minimal single number representation. The numbers 1978 and 22 are usedas examples.

MDCCCCLXXVIII XXII

To add the two numbers one simply throws them into a hypothetical bagand shakes as illustrated by FIG. 3. The bag itself is the variableassociation rule. It states that variables cannot wander off and thatexternal variables cannot intrude. Shaking the bag specifies that allvariables will eventually associate. There is no variabledifferentiation. Variables associate indiscriminately and there is noway to tell one variable from another inside the bag except by itsasserted value. The possibility expression is the set of value transformrules and the bag itself. The actuality expression is the values throwninto the bag.

The five Is will eventually get together and form the name IIIII. Thiswill invoke the value transform rule IIIII and the five Is willtransform into a V. There are then two Vs which will eventually make anX. The five Xs will make an L, the two Ls a C the five Cs a D andfinally the two Ds an M. What remains in the bag are two Ms. No moreinteractions are possible because there are no value transform rules forMs.

MDCCCCLXXVIII+XXII=MM.

There are a couple of difficulties with this expression. First, it isnot possible to determine when the expression inside the bag is fullyresolved either from inside or outside the bag. Second, some of thetransform rules require the confluence of five variables which mighttake a long time to occur. The Roman numeral system, however, was neverintended to be autonomously resolving as it assumed a rather intelligentinterpreter that could get all the proper variables together and couldtell when its resolution was done.

The following examples will demonstrate that fully determinableautonomous expressions can be expressed in isolated pure valueexpression environments. So that the reader can more easily follow theexamples the expressions from now on will be various forms of binaryinteger addition. The next example introduces the first form of thisexpression and is an expression whose transform rule names are nevermore than two values long.

Possible values are A,B,C,D,E

Transform rules are

AA [B]

BB [C]

CC [D]

DD [E]

This expression behaves similarly to the Roman numeral expression. If itis assumed that A=1, B=2, C=4, D=8, E=16 it can be seen that theseexpressions are equivalent to binary numbers. A number is represented byspecifying the appropriate digits. As with the Roman numerals there isno significance to spatial arrangement of the variables. Numbers areadded by throwing them in the bag and shaking.

DAC+BCA=EC; 13+7=20

This expression will complete more readily because no interactionrequires the confluence of more than two variables but there is still noway to tell when the computation is completed. For an expression toestablish its own completeness there must be a distinct lastinteraction. With the current expression there might not even be a firstinteraction.

AC+BD=ABCD

Combining these two numbers directly results in a minimally representednumber without any interactions at all. If no interactions occur therecan be no last interaction to indicate completion.

To guarantee the completeness of a progression of interactions there hasto be a guaranteed completeness to the form of the values entering intothe interactions. Completeness in this case means expressing thenonasserted place values as well as the asserted place values of thenumber. Another character will be attached to each value to indicate itsassertion or nonassertion. 1 means a value is asserted and 0 means it isnot asserted. So now AC would be represented as A1C1B0D0E0 or E0D0C1B0A1since order does not matter. It each two characters such as A1 or B0 isconsidered a single value then the number of distinct values has simplybeen increased to achieve more expressional differentiation. The 2character coding simply makes the meaning represented by each value moreobvious to the reader.

Each input number must now be represented by five values and it can beguaranteed that there will be an interaction for each possible placevalue whether that place value is asserted or not. This is a mild formof a variable association rule specifying that variables are alwayspresent in specific size groups and that each group has representativevalues from specific value groups. There is still, however, no orderrelationship imposed on the values.

For the new example the possible values are coded with two charactersfor convenience as follows:

Possible values are: E0,E1,D0,D1,C0,C1,B0,B1,A0,A1 Transform rules are:A0A0 [B0,A0] B0B0 [C0,B0] A0A1 [B0,A1] B0B1 [C0,B1] A1A0 [B0,A1] B1B0[C0,B1] A1A1 [B1,A0] B1B1 [C1,B0] C0C0 [D0,C0] D0D0 [E0,D0] C0C1 [D0,C1]D0D1 [E0,D1] C1C0 [D0,C1] D1D0 [E0,D1] C1C1 [D1,C0] D1D1 [E1,D0] E0E0[E0]   E0E1 [E1]   E1E0 [E1]   E1E1 [E0]  

The integer addition example now looks like:

E0 D1 C1 B0 A1+E0 D0 C1 B0 A1=E1 D0 C0 B0 A0; 13+7=20

To resolve the expression one still just throws all the variables into abag and shakes. As interact only with As, Bs with Bs, Cs with Cs, Dswith Ds and Es with Es. Each interaction generates a unique carry valuewhich will interact with its corresponding values until no moreinteractions are possible. If the two input numbers are represented asfive variables with each variable being a different one of the fiveflavors of value; Ax,Bx,Cx,Dx,Ex then the rules guarantee that theresult will be a similar number of five variables. The form of the inputname and result name is simply a convention among expressions.

Even with this completeness it is still not certain when the resolutionof the process is complete. The E rule will be invoked twice, once forthe input values and once for the carry value from the D rule. There isno way to tell which invocation is the last one.

To eliminate the ambiguity separate rules and values must be specifiedfor the input interactions and the carry interactions. Then there can bea definite last interaction which will be the carry into the Einteraction.

Because of minor combinatorial explosion in the possible values in thisexample they will be presented in two parts. The characters for thepositional flavor and positional assertion are represented by twoseparate tables. For instance the transform rule presented asAxAx[SBy,Az] when expanded in relation to the x x>>y z table reallyrepresents four separate transform rules: A0A0[SB0,A0], A0A1[SB0,A1],A1A0[SB0,A1] and A1A1[SB1,A0].

Possible values are: Dx,Cx,Bx,Ax UDx,UCx TDx,TCx,TBx SDx,SCx,SBxRDx,RCx,DONE  x = 0,1 Precedence of occurrence Transform rules are:relationships: AxAx [SBy,Az] 1 BxBx [SCy,TBz] 1 SBxTBx [RCy,Bz] 2 CxCx[SDy,TCz] 1 SCxRCx [UCz] 3 TCxUCx [RDy,Cz] 4 DxDx [TDz] 1 RDxSDx [UDz] 5TDxUDx [Dz,DONE] 6 for all rules: x x >> y z 0 0 >> 0 0 0 1 >> 0 1 10 >> 0 1 1 1 >> 1 0

The integer addition example now looks like:

E0 D1 C1 B0 A1+E0 D0 C1 B1 A1=E1 D0 C1 B0 A0 DONE

The new R, S, T and U values separately track the carry values and thecarry interactions. The DxDx rule is the input interaction and theTDxUDx rule is the carry propagation into D which is necessarily thelast interaction. The result looks just like the previous result exceptthat a new result value, DONE, is confidently generated by the lastinteraction.

All of the additional complexity of specification to achieve completecontrol is just a matter of more values and more value transforms thattrack intermediate values through the interactions to establish aconsistent and regular behavior that possesses a distinct lastinteraction. No new concepts or primitive elements needed to beintroduced. Control is not a primitive of process expression. It emergesfrom the defined expressional primitives properly arranged. It is simplyextra expression beyond what is required to just transform the data.

A familiar process has been completely expressed with full generality ina pure value environment. The expression is complete and self contained.Given the values, the transform rules and the variables it will proceedquite independently and autonomously in an orderly progression ofdistinct interactions leading to a distinct last interaction whichdetermines completion. There is no ambiguity in its behavior and itneeds no external assistance to effect its resolution.

E. Name Formation Dependency Relationships

How an input name is expressed and resolved depends on the resolutionresources available. The most critical resource for any expression isthe value transform rules available. If a very large number of Valuetransform rules with names eight values long were available then theprevious example expression could be resolved in a single interaction.The example used value transform rules with names only two values long.Consequently the expression could not be resolved in a singleinteraction.

An input name that is too long to be resolved in a single interactionmust be resolved a piece at a time by a necessarily dependentprogression of multiple interactions each resolving a smaller input namewhich is a piece of the larger input name. The eight value input namehad to be broken up into 4 separate input interactions. The results ofthese input interactions must be combined to form input names for innerinteractions and so on until the proper result values for the eightvalue input name are determined. The inner interactions depend on theresults of the input interactions to form their input names. Thisdependence of the formation of the input name for one interaction on theresult of one or more other interactions is name formation dependency.Name formation dependency relationships for value expressions areexpressed by value correspondence between result values and name values.

For a value expression this association of result to input nameformation is expressed by value correspondence. In the last exampleabove for instance the transform rule named RDxSDx is really 4 transformrules RD0SD0, RD0SD1, RD1SD0 and RD1SD1. Only one of these input nameswill be formed and the corresponding rule invoked. Which name is formeddepends on the CxCx rule which will result in SD0 or SD1 and the TCxUCxrule which will result in RD0 or RD1. The results of these twointeractions will form one of the four names of the RDxSDx rules. Withinthe context of the expression the values to form an RDxSDx name cannotcome from anywhere else but the resolution of a CxCx interaction and ofa TCxUCx interaction. The RDx of the RDxSDx transform me is a directassociation to the RDx result value of the CxCx transform rules.

The precedence of occurrence relationships for the example as shown inthe preceding TABLE and in FIG. 4 indicate the order in which theinteractions can occur. This order is determined by the input nameformation dependencies of the expression embodied in the transformrules. The 1 interactions can all occur simultaneously or at any time.Interaction 6 is guaranteed to be the last interaction. The input nameformation dependency relationships in the expression fully represent allthe possible concurrency. For instance the TCxTCx input name cannot beformed and the interaction will not occur before the CxCx and SCxRCxinteractions occur. Even though the CxCx interaction can occurimmediately the interactions dependent on it will not occur until CxCxoccurs no matter when that is. The control expressed in the expressionis complete and general. No matter how long it takes for eachinteraction to occur or how long it takes for the variables to gettogether, the expression will resolve correctly and completion ofresolution can be determined by the assertion of the DONE value. TheDONE value can perhaps open the bag and deliver the result.

This progression of interactions could resolve in a soup of just theeight input variables. Each interaction has two input values andproduces one or two result values. There will never be more that eightvalues asserted at any instant and there are only 5 result values. Thisview of resolution of the expression has the eight variables changingtheir asserted values as the transform names are formed. Two variablesget together, realize that they form a transform name and change theirasserted values to effect the transform. So the input variables are alsothe result variables. The following table illustrates the valuepopulations after each interaction stage.

Interaction stage 1 2 3 4 5 6 Ax Ax Ax Ax Ax Ax Ax Ax SBx Bx Bx Bx Bx BxBx TBx RCx Bx SCx SCx UCx Cx Cx Cx Cx TCx TCx TCx RDx Cx SDx SDx SDx SDxUDx Dx Dx TDx TDx TDx TDx TDx DONE Dx

Before the first interaction there are 4 interactable names. After thefirst interaction stage the only interactable name is SBxTBx. After thenext stage the only interactable name is RCxSCx and so on. The resultvalues are isolated from the input variables by the event of theinteraction. When AxAx interacts the input values disappear and theresult Ax appears. There are no more Axs and Ax does not enter into anyother transform name so it cannot interact further. When the DONE valueis generated there are one each of Ax, Bx, Cx, Dx and DONE laying aroundthat cannot interact further and these constitute the result values.

The interaction is the event that marks the progress of the resolution.The interaction forms a distinct event with a before and an after thatare isolated from each other because the input values and result valuesdo not exist simultaneously. Names form and disappear as they resolve.From each interaction new values arise to form new names which in theirturn interact, disappear and leave new values and new interactionpossibilities.

F. The Need for NULL Values

If one assumes conservation of variables, which is not necessary, thenof the eight input variables only five are needed to assert the resultvalues leaving three variables that are essentially expressional waste.These three variables must assert some value that is not relevant to theexpression. For instance if the RCxSCx interaction generated UCx andalso Bx the extra Bx would form a transform name with the result Bx orone of the input Bxs which in either case would mightily confuse theresolution of the expression. So these three variables must assertvalues different from the values of the expression. These will be calledNULL values because they are meaningless to the expression.

A NULL value for a value expression form is any value not specified inthe set of value transform rules. The example expression can resolve ina veritable sea of variables as long as all the other variables exceptthe eight input variables are asserting NULL values. The environmentthat an input name is formed in must be in an initial state in which allvalues of all proximate variables are NULL to the expression.

What is NULL to one value expression might be quite meaningful toanother value expression. So there could be many expression resolvingquite independently in a single frothing sea of variables.

G. Variable Expression

In the previous examples the differentiations and associations thatproduced an orderly computation were specified entirely in terms ofvalue differentiation and association by value transform rules. Thevariables themselves were indiscriminately associated so did notcontribute to the differentiation of the expression. They served as themedium of value assertion adequately available whenever needed. What isthe nature of an expression in terms of variables differentiated andassociated by variable association rules?

The variable association form of expression will be introduced byattempting to derive an expression directly from the last example valueexpression of four bit binary addition. The derivation will begin byassociating variables to mimic the input name formation dependencydiagram shown in FIG. 4 of the value expression. The same set of valuesand transform rules will be used initially. The diagram of FIG. 5 showsthe complete example of the candidate variable association expression.Each enclosed area is a variable. Areas whose boundaries touch areassociation relationships. The variable association rules are embodiedin the diagram and the diagram itself can be considered the expressionof the variable association rules.

Each value asserted during the resolution must have a variable to assertit. Each value asserted in the expression except for the input valuesmust result from an interaction. The variable that is to assert theresult value of an interaction must be able to see the formed input namefor that interaction. The variables must therefore be associated in sucha way that each result asserting variable is associated with thevariables that will form the input name to its interaction. The resultasserting variable is also considered to embody the transform rules ofthe particular interaction.

For instance the AxAx[SBy,Az] transform in the value expression used twovariables that simply changed values. For the variable expression itrequires four variables; two variables that assert the input values AxAxand two variables that assert the result values SBy and Az. Furthermorethe input name formed by the two input variables must be visible to bothof the result variables so the two input Ax variables have to beassociated with both the SB variable and the result A variable in afanout configuration.

The entire expression is derived in terms of these considerations. Thelabel of a variable in the diagram indicates the variable set it canassert. For instance the variable labeled TD can assert TD0 and TD1 andembodies the transform rules with TD result values. The transform ruleset AxAx[Az] is associated with the result A variable. The transformrule set AxAx[SBy] is associated with the SB variable.

The variables are no longer freely associating but are frozen into arigid structure by the association rules. The nature of interaction forthe variable expression is quite different from the nature ofinteraction for the value expression. The nature of the variables andvalues haven't changed but the nature of their possible relationships isdifferent. The example variable expression was straightforwardly derivedbut it will not work as it is presented. Discovering why this expressionwill not work and what is required to make it work will serve as anintroduction to the variable form of expression.

1. Continuously Interacting Structure

The variable association rules specify that the associated variables areinteractively proximate permanently and continuously. In the portion ofthe example shown in FIG. 6 the two input D variables are not associatedand cannot see each others values. The TD variable, however, can seeboth the input D values and respond to the input name formed by thevalues asserted by the two variables and assert the result value forthat transform name.

After TD recognizes a D,D input name and transforms its own value thevariable relationships do not change. TD is continuously seeing anyinput name formed by the two input Ds and it must continuously respondto that formed input name. TD cannot not assert a value. Nor can it justassert some nondescript value if it does not recognize a transform name.Its behavior must be completely defined so it must be always recognizinga valid input name and asserting a result value. Therefore there must bea set of value transform rules that span all the possible input namesthat can be formed by the input Ds.

Because a variable is always asserting a result value in relation to aformed input name, for a result value to be stably asserted the inputname seen by the asserting variable must be stably maintained. In otherwords the input Ds must maintain their asserted values if TD is tomaintain its asserted result value. TD's asserted result valuecontributes to an input name for another variable in the expression andmust itself be stably maintained. This continues through the entireexpression until all the variables of the expression have interacted andasserted the proper result values. So all the input values must bemaintained stably until the entire expression is resolved.

These resolution properties are quite different from those of a valueexpression. An interaction in a value expression occurs only when atransform name is formed. The input values that formed the input namedisappear and the result values appear marking a distinct progress eventin the resolution of the expression.. These new values are inherentlystably asserted by their variables until they form new input names,interact and themselves disappear. The value expression inherentlyresolves in a progression of discrete events ending with the assertionof the result values. The variable expression on the other hand iscontinuously resolving and asserting result values and this createsseveral difficulties with process expression.

2. Name Formation Ambiguity

The first problem with the continuous resolution nature of the variableexpression is that the example variable expression exhibits ambiguity ofinput name formation and direction of result propagation because of thereuse of a value set. The portion of the example shown in FIG. 7illustrates this ambiguity.

Variables do not posses any inherent directionality of interaction.Variable TD is associated with the input D variables as well as theresult D variable. Clearly there are input D values and result D valuesbut there is no way for the TD variable to know which variable a D valuecomes from. All it can see is three D values. Assume that the input nameasserted by the two input variables is D0D1 but the variable TD isseeing three D values instead of just two D values. The result Dvariable is having a backwards influence on a variable that recognizestwo value input names formed of D values. What result value should theTD variable assert when there are several simultaneously valid two valueD input names visible to it?

The D variables have been referred to as input and result because thatwas their role as mapped from the pure value expression where they wereindeed unambiguous input and result values. Because of this discreteevent nature of the value expression the input D values did not getmixed up with the result D value. As far as each variable in thevariable expression is concerned it is still in a pure value environmentlooking for a input name in a sea of values. The sea for each variableis now a small stagnant puddle but the variable doesn't know that.Variable interaction is completely directionless. Therefore whenvariables are associated there is no inherent input end and result end.So the influence of name formation in a variable expression propagatesthrough variables in all directions at once.

This is why the example expression shown in FIG. 5 will not work. Theresult values can get confused with the input values in name formationand asserted values can influence interactions backwards in theexpression. This is not an invalid form of expression. A molecule forinstance or a gravitational system are variable association expressionsthat are formed with continuous interaction pulsing through theirstructure in all directions simultaneously. This discussion, however, ispursuing process expressions that proceed in a more or less orderlymanner to a more or less definite result. So strict directionality ofinteraction influence must be imposed on the variable expression.

3. Need for Value Isolation

Referring to FIG. 7, the only way to establish directionality within thedefined model for a variable expression is to differentiate the inputfrom the result with different values. To establish directionality allvariables directly associated must assert different value sets. In FIG.8 it can be seen that variables 1 and 3 cannot assert the same value setwithout confusing variable 2. Only 1 and 4 can assert the same value setwithout forming ambiguous input names for 2 or 3. So there must be atleast three sets of value transform rules with nonintersecting valuesets to directionalize a variable expression and variables assertingidentical value sets must be at least three associations apart.

It can be seen in FIG. 5 that several variables violate this rule. It isobvious that the result values can get mixed up with the input valuesthrough the variables TB, TC and TD. The only way to fix this ambiguityis to isolate the assertions of identical value sets by inserting extrabuffering variables into the expression that assert different valuesets.

This may seem a complex requirement to impose on an expression but theproblem in general is quite simply solved. By adding enough variables toan expression and choosing two value sets that the process expressionproper does not use it can be assured that there are always twovariables between each variable asserting a process expression propervalue and hence the use of any identical value set by the processexpression proper will always be three variable associations apart.These two inbetween variables with their value sets can form a standarddirectional interconnection unit through which all process expressionproper variable associations are formed. This standard interconnectionunit will be called an interaction locus.

H. The Interaction Locus

The interaction locus isolates its input value associations from itsresult value associations and establishes directionality of interactionfor variable expressions. It doesn't matter what value sets are usedinside the locus as long as they are different from the value sets usedby the process expression proper. Now that all process expression propervariables and values are isolated such that identical process expressionproper values will not lead to input name formation ambiguities theprocess expression proper can be rendered entirely in terms of a singlevalue set.

FIG. 9 illustrates the role of the interaction locus. Variables 3 and 4are the isolation variables. Variables 3, 4 and 5 are the interactionlocus. Variables 1, 2 and 5 are process expression proper variables. 1and 2 are the input variables and 5 is the result variable. Variables 1,2 and 5 will assert the value set 0, 1. Variable 3 will assert the valueset X, Y. Variable 4 will assert the value set S, T. The following valuetransform rules sets will be associated with each variable.

Variable Variable Variables 3 4 1,2,5 00[X] X[S] S[0] 01[Y] Y[T] T[1]10[Y] 11[X]

Variable 3 only recognizes input names of 0, 1 and asserts result valuesX, Y. It embodies the value transform rules for the interaction.Variable 4 only recognizes input names X, Y, and asserts result valuesS, T. Variables 1, 2, and 3 only recognize input names S,T and assertresult values 0, 1. An x assert by variable 3 will result in a 0asserted by variable 5. A y asserted by variable 3 will result in a 1asserted by variable 5. The interaction locus establishes thedirectionality of influence and isolates the input values from theresult values. The process expression proper variables 1, 2 and 5 allrecognize and assert the same value set but their values never get mixedup because they are all isolated from each other by the isolatingvariables of the interaction locus. The process expression proper cannow be entirely expressed in terms of a single value set in terms ofassociated interaction loci. The interaction locus recognizes the valueset 0, 1 as input values and asserts the value set 0, 1 as resultvalues.

Different transform sets can be assigned to variable 3 of theinteraction locus to provide interaction loci with different nametransformation functions. Variable 4 is just a buffer variable alwaysassociated with variable 3 and variable 5 is the result assertionvariable always associated with variable 4 so the three variables can beconsidered as a single unit of expression. The expression unit can beidentified by the transform rule set associated with it. The inputvariables 1 and 2 are the result variables of some other interactionlocus. The new directionalized expression of an interaction locus mightlook something like the expression shown in FIG. 10.

T1 represents the isolation variables and the transform rule setassociated with the interaction locus and part of the interaction locusthat will receive and recognize the formed input names. R represents theresult assertion part of the interaction locus. T1 is the input end andR is the output end. An expression of associated loci might look likethe expression shown in FIG. 11. The elements with I are the inputvariables asserting the input name to the expression.

A little graphic stylizing will provide a more familiar look to theexpression as illustrated in FIG. 12. Each interaction locus transformset can be represented by a different shape and that shape canexplicitly indicate the direction of interaction. With the convention ofthe interaction locus a variable association expression can be viewed inthe familiar terms of interconnected transform elements or functionelements, such as logic gates or members.

The interaction locus establishes directionality of interaction with aconvenient and uniform convention by isolating the input values from theresult values of an interaction. A directionalized variable expressioncan be rendered without interaction loci. One could very carefullyassociate variables and assign values and transform rule sets so thatthe isolation criteria is satisfied. Its just more involved. Variableexpressions also do not have to be directionalized.

A Bounding Convention

Directionalized interaction, however, is critical for extensiblecomputation. Data value resolutions such as arithmetic could not existwithout directionalized interaction. Real world examples include theelectromagnetic switch, the electronic tube and the transistor. For theelectromagnetic switch the input current value influences a magneticfield value which influences the physical position value of the switchwhich influences the result current values. Identical input and resultvalue domains are isolated by different value domains just like theinteraction locus derived within the model. The tube and transistorsimilarly use different physical interaction domains to isolate theinput from the output. A current value on an input wire influences acharge value in a vacuum or semiconductor which influences electron flowin the same medium which influences the current in the result wire.

The interaction locus is a bounding convention. It encapsulates anexpression that can be quite arbitrarily represented itself but whichpresents a specific convention of interactive behavior to all otherinteractive elements participating in the convention. This boundingconvention establishes the first instance of what might be called fromone viewpoint an imposed expressional abstraction or from anotherviewpoint an emergent expressional facility. In either case itestablishes a uniformity and regularity that makes the wholeconsiderably more than the sum of its parts.

If one looks at a variable expression variable by variable as justassociated variables the interaction loci might not be at all evident.There is no guarantee in any expression that all the interaction locilook the same. The only requirement is that the boundaries look the sameto each other. Their insides may vary dramatically.

In a modern computer for example the transistor circuits that implementa logic gate or information processing member may vary dramaticallybetween chips made by different manufacturers. Some interaction locihave magnetic values inside and some have mechanical values inside.Imagine an expression of a processor made from chips from differentmanufacturers expressed solely in terms of transistors, capacitors andresistors with no clue as to where the boundaries of logic valueexpression were or that logic value expression had anything to do withthe expression. Without the overlay of logic gates on the expression itwould be just a huge network of electronic elements and extremelydifficult to understand.

But the overlay is exactly that. There is nothing intrinsically “real”about it. It is just a convention that must be maintained amongconsenting expressional elements. The interaction locus is an imposedregularity and uniformity that imparts an abstract level of meaning tothe expression that the individual elements can know nothing about andcannot anticipate the existence of. It shows an expression of complexmeaning the possibility of which could not be projected from the natureof the variables and values themselves.

I. Directly Associable Interaction Loci

The example expression can now be rendered in terms of theseinterconnected interaction loci. The following three interaction locitransform sets will be defined and will be called ADD, CARRY and DONE.Ignore for the moment that DONE is nondiscriminative and always assertsa 1.

ADD CARRY DONE 00[0] 00[0] 00[1] 01[1] 01[0] 01[1] 10[1] 10[0] 10[1]11[0] 11[1] 11[1]

The example process of four bit binary addition begun in FIG. 5 can nowbe constructed entirely in terms of the value set 0, 1 and theinterconnection of these three types of interaction loci as shown inFIG. 13. The ADD locus is A. The CARRY locus is C. The DONE locus is D.

The variable type labels are attached to the interconnection variablesin the example to show this example's correspondence with the previousexamples. It will be seen that there is a one to one correspondencebetween the variables of the initial variable expression example andthis example. What were directly associated variables in the initialexample are now directly associated interaction loci. It will beconvenient to continue referring to the connecting transmission paths asvariables that associate the interaction loci.

In the example names are always formed and interaction activity isoccurring continuously. There are no necessary and discrete interactionevents. If several transform rules in an expression specify the sameresult value then it is possible for a new input name to form and nochange event to occur in the expression. For example if the input Dschange their formed name from a name that specified the same resultvalue then TD will continue asserting the same result value and no eventwill have occurred as a result of the newly formed input name. Thisraises the problem of determining when a resolution of an input name hascompleted. The problem of completion reveals a number of furtherproblems with variable expressions.

The first difficulty is that a DONE value is always asserted. The inputvalues to the DONE variable cannot be differentiated. Both UD and TD maybe either 1 or 0. The second difficulty is that even if a DONE conditioncould be determined from UD and TD it cannot be guaranteed that theresult D value will be asserted before or simultaneously with the DONEvalue. In the value expression they were both generated by a singleinteraction but in the variable expression they are asserted bydifferent interactions that may resolve at different speeds. So it ispossible that the DONE value could be generated before the result Dvalue. So the issue of resolution completion is still indeterminate forthis expression.

J. The Need for a NULL Value in Variable Expressions

There is no way for an expressional element to become meaningless withinthe context of a variable expression. All variables are constantlyassociated and all values are constantly forming valid input names andinteracting. Elements of a value expression, on the other hand, candisappear from the expression and become meaningless. A variable thatwas participating in the resolution can suddenly assert some NULL valuethat is not part of a transform name of the expression and becomemeaningless to the expression. Elements of a variable expression cannotjust drop out of the expression. The variables are locked in a rigidassociation structure and any value that they assert is inevitablyinfluential in the expression and the expression must account for them.Therefore every possible formed input name must be accounted for by atransform rule. Any NULL value in a variable expression must be includedin the transform rule names and be integral to the expression. So for avariable expression there must be a specific value assigned to meanNULL.

The introduction of a NULL value can resolve the problem of resolutioncompletion. The NULL value essentially allows a variable to expressmeaninglessness within the structure of associations. The basic strategyfor the transform rules is to specify a NULL result value if their inputname includes a NULL value. In this manner formed input names can berecognized as valid or invalid. A valid input name is one with no NULLvalues. So although there are always input names formed and recognized,an invalid input name can suddenly become a valid input name providing adistinct resolution progress event. The variable recognizing thissuddenly valid input name will transform its asserted result value fromNULL to a nonNULL result value.

The following set of transform rules applied to the example expressionwill allow the assertion of the DONE value to be a distinct event in theresolution.

ADD CARRY DONE 00[0] 00[0] 00[1] 01[1] 01[0] 01[1] 10[1] 10[0] 10[1]11[0] 11[1] 11[1]

anyNULL[NULL] anyNULL[NULL] NULL[NULL]

The expression must begin with all asserted values in the expressionNULL including the input values. For the value expression thisrequirement was simply that no values defined in the expression wereasserted in the variable soup. If there were any defined valuesinitially asserted besides the input name values the resolution would beconfused. The variable expression has the same problem. If any of thevariables in the expression are already asserting nonNULL result valueswhen the input name is applied to the input variables the resolutionwill be confused. Therefore all the variables in the entire expressionmust be asserting NULL values when the valid input name values areapplied.

It is this initial state of NULL that insures the occurrence ofprogressive interaction events. As each input name is recognized by eachinteraction locus the asserted NULL values will change to meaningfulnonNULL values in an orderly progression of value transforms propagatingthrough the expression until all the result values are valid. Theexpression must be reinitialized to NULL before another input nameresolution can be started. This can be accomplished by simply presentinga NULL input name. The NULL result values will propagate through theexpression just as the valid result values did. So the NULL conventionrequires that there be an alternating cycle of valid input names withNULL input names.

With the above transform rule sets and with the expression in an initialNULL state with the input values NULL and all interaction loci assertingNULL values The DONE variable can make a distinct transition from NULLto nonNULL. The entire expression will continue to assert NULL values aslong as the input name values are NULL. As the input name values becomenonNULL the interaction loci begin to assert valid result values. Whenthe two input As are valid the result A and SB values will become valid.The result B cannot become valid until the input Bs are valid.

Since all the result values are dependent on the formation of the inputname the result name will not be fully valid until the input name isfully valid. For instance both input Ds can be valid but if one of theother input values is not valid then the result D will remain NULLawaiting a valid carry value. The result D will not become valid untilRD and UD become valid. D will be the last result value to be generatedand DONE relying on the same input name can assert the completion of theresolution by becoming nonNULL. The assertion of the DONE value is now adistinct completion event but there are still the race conditions thatwill allow the possibility of DONE being asserted before the valid D isasserted. The solution to this requires some reorganization of theexpression as shown in FIG. 14.

The result of the DONE variable is now directly dependent on thevalidity of all of the result values of the expression. A valid DONEvalue will not be asserted until after all the result values are validlyasserted. The expression can now autonomously assert its own completion.With the NULL convention and the interaction locus convention this isfinally a variable association expression of the example 4 bit binaryaddition process that will work.

The NULL convention scales up through combinations of interaction locito endow a large expression of associated loci with the same behavior asa single locus. The large expression will only express a completelyvalid result name when a completely valid input name is present. Becauseno single locus changes its result from NULL until a valid input ispresent there are no race conditions. The result values will propogatethrough an expression in an orderly wavefront of valid result valueswith no invalid spurious values asserted anywhere in the expression atany stage of the resolution and finally a valid result name for theexpression is asserted. The concurrency of the expression is fully andreliably managed. When a result value goes nonNULL it is asserting avalid result of a valid input name.

This final example is a familiar form of expression which can be viewedas interconnected processing elements or functional elements. Theconfiguration of associated interaction loci is the possibilityexpression. The input values presented to the input of the configurationis the actuality expression It is still just variables associated byvariable association rules asserting values that form input names thatinteract and transform according to value transform rules. The variableexpression relies on the same primitive conceptual elements that thevalue expression relies on.

K. Summary

A view of process was presented which characterized process in terms ofchange occurring to existential entities due to interactions among thoseentities. Interactions occur in the context of a configuration ofpossible interactions. Which possible interactions will occur isdetermined at interaction time by the appearance of a specificationwhich determines which possible interactions will proceed. The possibleinteractions are specified by a possibility expression and the actuallyoccurring interactions are determined by an actuality expression thatappears at the time of interaction.

The invocation model of process expression was introduced as aconceptual accounting for general process expression. It consists of twoprimitive elements, variables and values, and two forms of compositionrules for these elements, variable association rules and value transformrules. The variables and values are the existential elements of themodel. Variables are pure existential entities. Every variable is alwaysasserting a value which determines its interaction possibilities thatcan change during interaction.

Variable association rules specify which variables are interactivelyproximate and hence interactable. Value association rules specify whichvalues are interactively proximate and can interact and also what resultvalues will be asserted when the proximate values interact.

These primitive elements can not only specify individual interactionsbut they can specify compositions of multiple interactions to express alarger process as a progression of interactions in terms of dependencyrelationships among result values and the values of actualityexpressions (input names). These dependency relationships can beexpressed two ways within the model. They can be expressed ascorrespondence between combinations of result values and names of valuetransform rules or they can be expressed as association relationshipsamong variables by variable association rules. This means that there aretwo distinct but inextricable realms of expression within the invocationmodel; the value expression form and the variable expression form. It isthe relationship between these two realms of expression that relate manyforms of expression generally thought to be quite distinct andunrelated. Of these two realms one seems to be more fundamental than theother.

In a value expression differentiation and association of meaning withinthe expression is almost entirely in terms value transform rules.Variables are not explicitly differentiated and consequently cannot beexplicitly associated. They are either all constantly associated or areindiscriminantly associating. Interactions occur as interactable namesare formed.

A value expression can stand alone as an independent autonomousexpression on the basis of the primitive definitions of the model. Aninteraction in a value expression is a distinct resolution progressevent. The input values disappear and the result values appear. Aninteraction is inherently directional because the result values cannotbe confused with the input values. Result values are independentlymaintained by their asserting variables from interaction to interaction.An expression resolves in an orderly progression of interactions andunambiguously expresses its own completion by the existence of theresult values.

In a variable expression differentiation and association of meaningwithin the expression is almost entirely in terms variable associationrules. There must be values but all possible names are interactable sothere can be no differentiation and specific association among thevalues. They are all constantly interactively proximate. Interactionsoccur wherever variables are associated.

The variable expression form on the other hand, requires a liberal doseof convention for it to be an autonomous expression with the sameexpressional qualities that are inherent for the value expression form.It required the NULL value convention to provide distinct interactionevents. It required the interaction locus bounding convention to avoidinput name formation ambiguity and establish directionality ofresolution progress. These conventions were, however, definable from theprimitive elements of the invocation model. No new primitives needed tobe introduced.

It is all just associated variables asserting values that interactaccording to value transform rules in a dependent progression of inputname formation and resolution. The invocation model of processexpression has been introduced as a conceptual foundation forconsidering process expression of both natural and artificial processes.The implication is that there is no fundamental difference betweennatural and artificial expressions. Humans do it in much the same waythat nature does it. Process expression far from being an artificialundertaking that can be arbitrarily adjusted to fit any desiredconceptual model is found to have inherent limitations and necessaryrelationships much like the hard physical sciences.

All forms of process expression are related by the necessity todifferentiate existential elements and express changes of existentialdifferentiation through interactive associations of those elements. Allprocesses resolve in a dependent progression of actual interactionswithin a context of possible interactions.

III. The Process Expression Landscape

This discussion explores the nature of process expression with varyingquantities of value differentiation and variable differentiation. Thediscussion will begin with an arbitrarily defined baseline exampleprocess presented as a pure value expression. The discussion willprogress through several forms of process expressions all representingthe same example baseline process in different forms. Finally the purevariable form of process expression will be derived.

A. The Baseline Example Process

The example process for this discussion consists of 15 existentialdistinctions 6 of which can interact in 9 possible ways producing one of9 possible results. A pure value expression is used as the baselineexpression of the process because it is straightforward and intuitivelyunderstandable. The 15 existential distinctions in the process aredifferentiated with 15 unique values. All 9 possible interactions amongthe 15 distinctions are specified in terms of value transform rules.

Tables will be used in this section to represent sets of value transformrules because they are more compact and more convenient to read than thecharacter string representations. It should be remembered that the tableis not a single transform rule but just a convenient presentation of aset of transform rules. The corresponding character string expressionsof the transform rules for the baseline example are also presented forthis example to show the correspondence of the two forms ofrepresentation. The baseline example process was defined by simplyfilling in the table with arbitrary values.

All the existential differentiation in the process is expressed byunique values. There is no case where the same value expresses twodistinctions. The set of value transform rules is a complete expressionof all the interaction possibilities among the 15 distinctions. The purevalue representation uses transform rules with input names two valueslong and will resolve in a single interaction step.

B. Limited Values

The discussion will begin with the possibility that there are not enoughvalues to directly represent all the distinctions of the example.Suppose for instance that there are only 4 values available I, J, K andL to represent the 15 distinctions of the process. The expression of thedistinctions will have to be encoded by using multiple values torepresent each distinction previously represented by a single uniquevalue. An arbitrary assignment of encodings might be as follows:

As a result of the encoding identical values are now used to representmore than one distinction. For instance the input names IJ and JIrepresent two different distinctions so the Is and Js of each input namemust be differentiated. This can only be done by differentiating amongthe variables expressing the values. This J means something differentfrom that J because it is expressed by a different variable with aspecific association relationship to the variable of the second J.Differentiation lost by limiting value differentiation must be made upby differentiation of variables. The encoded input name must beexpressed by 4 differentiated variables. The 4 variables cannot beinteractively proximate.

Even though identical values are differentiated by different variableswhen these variables are associated at an interaction place thevariables become interactively proximate and loose their differentiationand the identical values lose their differentiation. The interactionplace of interest for this discussion is the interaction-locus definedin the previous chapter and in this respect it is like any other placeof interaction. The interaction locus cannot discriminate that thisvalue came from the first variable and that value came from the secondvariable. Inside the interaction locus there is a collection of valuesjust like a pure value representation and some of these are identicalvalues. For instance if the input name to a locus were IJJI the locuscan only determine that there are 2 Is and 2 Js. The input name couldeasily have been IIJJ. At the interaction locus all ordering is lost andonly quantities of values can be discriminated. Therefore an arbitraryencoded input name cannot be unambiguously discriminated in a singleinteraction in a single interaction locus. In fact the only form ofencoded input name that can be unambiguously discriminated by a singleinteraction locus is an input name with all values identical. For thecurrent example the only unambiguously discriminable input names areIIII, JJJJ, KKKK and LLLL. There is only one possible input name with 4Is only one with 4 Js and so on.

Furthermore an interaction locus can generate only one value for oneresult variable. Since the values in the locus are not differentiatedthe locus cannot decide that this value goes to the left result variableand the other value goes to the right result variable. Any number ofinput variables can be mixed into the locus but the internal mix cannotbe unmixed to several result variables. This in itself further limitsthe discrimination possible in an interaction locus. If there are onlyfour possible values that the result variable can assert then only fourdistinct input names can be discriminated by the locus.

An interaction locus with 4 input variables for the current example canonly discriminate 4 unambiguous input names and can assert only 4 resultvalues. A locus with only two input variables can still unambiguouslydiscriminate only 4 input names; II, JJ, KK and LL and assert 4 resultvalues so in general there is a rapidly diminishing return ofexpressiveness for associating more than 2 input variables to aninteraction locus. All of the examples of this discussion will assume 2input interaction loci.

The discrimination power of the interaction locus is considerably lessthan the expressivity of its possible input names. The encoded inputname clearly cannot be resolved in a single interaction locus. A singleinteraction locus can only receive a small piece of the encoded inputname. There must be a coordinated cooperation of many interaction locito represent the complete example process with each locus providing apartial resolution of a small piece of the input name. It is thisprogression of partial resolution results that determines the input nameformation dependencies among the interaction loci and provides thestructure of the expression.

Before continuing it will be remembered that for a variable expressionto be autonomous one of the values must be assigned the NULL meaning.This would leave only 3 of the 4 values representing process propervalues or a fifth value would have to be added to represent 4 processproper values. Having mentioned the need for the NULL value the issuewill be ignored for the bulk of this section and only reintroduced atthe end to establish the nature of the pure variable expression. Thefollowing discussion will sound more familiar in relation to currentpractice and experience if the NULL value issue is ignored. Traditionalforms of representation deal only with representation of process properdistinction and control is introduced as carefully coordinated externalexpressions such as system clocks, delay lines and other timed events.

The variable association expression will be presented as a directedgraph. A node is an interaction locus. The arcs are the associatingvariables. A spanning set of transform rules is associated with eachlocus that will resolve all the possible input names.

Since the resolution duties that each locus can perform are very limitedthe expression must be a coordinated progression of interaction loci.What is the fundamental rationale of forming such an expression? How arevalue transform rules assigned at each locus? How are the input nameformation dependency relationships determined? The pure value expressionwas direct and intuitive but the encoded expression is neither directnor obvious. It was easy to specify that A and Z go to 5 but how can itbe specified that IJ and JJ go to U with the tools at hand?

There is no particular pattern to the example encoded table so the onlygeneral way to approach the expression is to recognize each possibleinput name and generate the appropriate result names. Each input namecan be recognized individually and that recognition expressed bydifferentiated variables. The correct set of result values can then beasserted and these values collected to a single set of result variables.

The first stage of resolution is to recognize the input names. It hasalready been shown that a single interaction locus cannot discriminatebetween the input names IJ-JI and II-JJ so it must take multiple locijust to discriminate input names. Can a single locus discriminate justIJ from all other input names? The answer is again no because the inputname JI might be a valid input name. The only possibility is totransform the expected input name into an unambiguous standardrecognition name with 2 identical values and discriminate this standardrecognition name.

So the first task for an interaction locus is to transform singlevalues. This can be done with appropriate explicit transform rules ateach locus or it can be done with a more general rotation locus whichcan be applied multiple times to get the desired transform. The valuescan be put in some circular sequence and the rotation locus simplystates that each value is transformed into the neighbor in a particulardirection. Any value can be transformed into any other value through theappropriate number of loci. The example rotate locus is shown in thetable below. A rotate locus will be graphically represented as a smallsquare with an R in it as shown in FIG. 15.

The standard recognition name will be chosen to be LL. To recognize aspecific input name the input values will be rotated such that thespecific input name will be rotated to LL. If LL is not the result ofthe rotation then the input name was not the specific input name to berecognized.

The rotate configuration shown in FIG. 15 will set up the input name IJfor recognition. If the input name presented on variables A and B is IJthen the result of the rotates will be the name LL. If the result nameis not LL then the input name was not IJ.

The next task for an interaction locus is to discriminate LL from allother input names. The equality locus recognizes exactly one input name.The example equality locus is shown in the table below. An equalitylocus will be graphically represented as a rectangle with an = signinside as shown in FIG. 16.

This set of transform rules generate an L for the input name LL and an Ifor all other input names. The table is representing a control orlogical significance and has established the convention that I meansFALSE and that L means TRUE.

Since an interaction locus cannot discriminate order among values andcannot tell IJ from JI. All value pairs with mirror symmetry mustresolve to the same result value. Therefore all tables of transformrules that use identical value sets for both input variables mustexhibit diagonal symmetry. The only input names that an interactionlocus can unambiguously recognize are the names along the diagonal forwhich both values are the same.

With these two interaction locus transform rule sets all the input namescan be recognized. Each of the 9 possible input names is individuallyrecognized by a separate group of interaction loci forming a recognitionstage. The input variables are just routed to each of the input namerecognition stages. One of the stages will recognize an input name andits result value will be L (TRUE). All the rest of the stages will havefailed and their result will be I (FALSE).

For the example expression the input variables are labeled A, B, C and Dand the result variables are labeled X and Y as shown in the tablebelow.

FIG. 16 shows the input name recognition section for the exampleexpression. In the example there are 9 possible input names of fourvalues each. Since a single locus can only recognize two value inputnames the input name recognition is broken into two stages in whichinput names of 2 values are recognized and then these recognitionscombined to recognize the 9 four value input names. The recognitionitself is represented by nine distinct variables. Only one variable at atime can assert TRUE while the rest must assert FALSE. Of course theycan all be false if the input name matches none of the recognized inputnames.

The single assertion of TRUE from one recognition stage is used togenerate the result name associated with the recognized input name. Thenext task for an interaction locus is to assert a particular value ifenabled by a TRUE value and to assert a default result value if notenabled. The transform set shown in the following table implements theassertion locus. An assertion locus will be graphically represented by arectangle with an A inside as shown in FIG. 17.

One input variable of the assertion locus is set constantly to thedesired value an the other input is the enable variable which will be Lor 1. If it is L then the constant value will be asserted on the resultvariable. If the enable input value is I then the default value of Iwill be asserted on the result variable. Choosing I as the defaultresult value is an arbitrary convention that all other loci must relateto once it is established. It is generally convenient to choose thedefault result value to be the same as the FALSE value.

So all the assertion loci except one will assert I and the selected onewill assert a nondefault value which may also be I. All these valuesmust then be combined through a priority network such that anynondefault value overrides all the default values and is asserted by theresult variable for the whole expression. This prioritized collection ofasserted values is the last general duty of an interaction locus. Eithertransform set shown in the following table will implement a prioritylocus. A priority locus will be represented graphically by a rectanglewith a P inside as shown in FIG. 17.

At each of these loci either two Is will be presented or one I andanother value. In all cases the other value will be asserted on theresult variable of the locus, and will make its way through a tree ofsuch loci to the result variable of the whole expression.

FIG. 17 shows the entire example expression to assert the value for theY result variable including value generation and prioritized collection.The 9 assertion loci with one input set to a constant generate the Yresult value for the recognized input name. The asserted value is thendirected to the result variable via the tree of Priority collectionloci. The Y result value is shown because it takes on all 4 values andmakes a better example than the X result. The X result value isgenerated by a similar assertion and priority network driven in the sameway by the same input name recognition TRUTH variables.

The expression can be viewed as consisting of two halves. The first halfrecognizes the input names. The second half asserts the proper resultvalue on the result variables. Looked at another way the expression canbe viewed in three parts. There is an input section that relatesdirectly to the input values and recognizes pieces of the input name.There is an internal section strictly in terms of logic values whichcombines the partial recognition pieces recognize the larger input name.Then there is the result section which asserts the result values basedon the logic values of the internal section.

This initial example provides a convenient context within which manyother issues of process expression can be discussed.

C. Optimization

This expression can be optimized in several straightforward ways. In theinput name recognition stage, redundant rotations can be eliminated. Ifthe same rotation is applied to the same variable more than once theresult of a single rotation stage can be fanned out to accommodate theother inputs as shown by FIG. 18.

Referring to FIG. 19, in the result generation stage the logical controlvalues do not have to be asserted as constants because the logic controlvariable is already L or I. The L can just be passed through. Is do nothave to be passed through at all because I is the default value. If nohigher priority value is selected the result will be I.

It will be noticed that only J and K are specifically generated forresult values. The recognition stages that generate I value results forthe Y variable are still retained to control the result generation stagefor the X result variable. If there were an II result value therecognition stage for this input name could be eliminated entirely.

There is one more straightforward optimization that can be applied toeliminate redundant assertion loci. The logic values can be collectedfor each result value before asserting that result value. This allowsthe use of only one assertion locus per result value. The finaloptimized expression is shown in FIG. 20.

There are likely many other techniques that could be developed toachieve other optimizations. For instance, a Kamough-map-like techniquemight be possible to determine what minimal set of input names actuallydetermine the result values. Such a technique might have shown that thearbitrary encoding left the A variable as a constant and that the inputnames might possibly be discriminated with only three variables.

D. Relation to Binary Logic Representations

This expression corresponds directly to the familiar binary circuitexpressions. The rotate locus corresponds to the NOT gate. The equalitylocus corresponds to the AND gate. The priority collect locuscorresponds to the OR gate. There is no counterpart of the assertionlocus because binary expression does not have any intermediatenonlogical values to assert.

The expression of a binary circuit follows the same strategy as the fourvalue example expression. It recognizes input names and generates aunique result for each unique input name. The binary expression howeverhas some unique advantages over a multivalue expression. There iscomplete intersection between its logic values and its expression propervalues. If 0 is chosen as the default result value then any input namethat results in all 0s need not even be recognized. Only input namesthat result in a 1 need be recognized. So the basic strategy for binarycircuits is to recognize all input names that result in a 1. If 1 isassigned as the internal TRUE value the truth of the recognition is thedirect result value. All the truth values from the recognition stagesare collected and that is the result of the expression. This can beillustrated with the half adder circuit. The truth table for thehalfadder circuit is shown below and the circuit itself is shown in FIG.12.

There are only two input names 01 and 10 that generate a 1 for the Sresult variable an only one input name 11 that generates a 1 for the Cresult variable. These three input names are recognized by rotating theinput names to the standard equality name (11) and determining equalitywith an AND gate. The truth value (1) of their recognition is collectedthrough the OR gate as the result value. If none of these input namesare recognized the FALSE value (0) of the recognizers will be assertedas the default result value. The binary logic circuit representation isconstructed with exactly the same representation principles as the fourvalue example expression.

E. Definable Transform Sets

Can the example process be expressed more directly with four values? Canthere for instance be a more direct mapping from the input names to theresult names without going through the internal logic? It seems aparticularly inefficient use of expressive resources to have a locusassert only two result values when there are four possible resultvalues.

This would mean that each recognition locus would have to discriminatethree or four input names instead of just two input names. Since asingle locus can only discriminate four input names and not all inputnames can be recognized by a single locus, one of the result values mustbe assigned a meaning of FALSE. So each locus can recognize only threepossible input names. The only input names that a locus canunambiguously recognize are the input names with identical values II,JJ, KK and LL. If I is assigned to be the FALSE value then any three ofthe four unambiguous input names can be discriminated with the resultvalues J, K and L. These names then are the standard recognition namesfor a locus and any input name can be recognized by rotating it to oneof these recognition names.

A single locus cannot however recognize three arbitrary input names.Only one rotation can be applied to each of the two input variables of alocus so the three input names recognized by a single locus must berotation neighbors. The same rotation must rotate one input name to JJthe second to KK and the third to LL. For example the input names IJ, JKand KL can be recognized by a single locus by applying a single rotateto the first variable. The following transform set will unambiguouslyrecognize three input names that are rotation neighbors.

The internal loci are now presented with all possible combinations ofthe four values instead of the two logical values as in the firstexample. The internal expression stage is no longer logical. In shortthe internal loci are faced with the same name recognition problem thatthe input loci are presented with. Now however there are only twovariables instead of four variables. The same rotation neighbor strategycan be applied to these internal names so that a single locus canrecognize more than one input name.

The transform rules for each locus can be custom defined to assert thedesired result value for the recognized input name directly. Assume thefollowing input name to result value mapping JJ→K, KK→J and LL→J. thefollowing locus will generate the appropriate result values directly.

Other rotationally related groups of input names could be accommodatedwith similar custom loci. There must still be several of these stagesand their result values must still be priority collected to the finalresult variables. So some advantage in the number of variables and locican be gained with custom transform sets defined for specific loci.

More advantage can be gained by carefully encoding input names insteadof arbitrarily assigning input names. FIG. 21b shows such an encoding tooptimize the expression. In this example rotational neighbors areassigned to each of the two input variables. FIG. 21a shows the graphicrepresentation of a custom interaction locus. The 3 letters in therectangle indicate the setting of the diagonal values in the transformtable. The complete optimized expression is shown in FIG. 21c.

Differentiation resources have not heretofore been fully utilized. Sincethe variables AB and CD are differentiated by variables they do not needto be differentiated with unique values and can be assigned the sameencoding input names with no ambiguity. The input distinctions can bedifferentiated with two variables with three values each as shown inFIG. 22a. Four variables are not needed to differentiate the inputdistinctions. So the encoding can be improved even more and a wholeexpression stage eliminated as shown in FIG. 22b.

For this particular expression the third recognition locus results inonly Is so it can be eliminated along with one priority locus. Apartfrom that, this is probably close-to the minimal form of expression ofthe example process achievable with four values.

Whether a expression is rendered in terms of a few standard loci or interms of custom definable loci is a matter of choice, possibility andpracticality. The fancy interaction loci discussed here may not bepossible in most practical expression environments. For instance it mayonly be possible to detect a threshold presence of a single value.Traditional electronic logic gates only recognize a voltage threshold.

F. More Available Values

If more values are available, for example 6 values, the input names canbe expressed without replication of values. This eliminates theambiguity inside the interaction locus that limited input namerecognition to input names of identical value. There no longer need beidentical values presented to an interaction locus. The nine resultdistinctions must, however, still be encoded with two variables as shownin FIG. 23a. The entire example process with both X and Y resultvariables can now be expressed With two custom defined interaction locias shown in FIG. 23b.

If 9 values are available the result values need no longer be encoded intwo variables as shown in FIG. 24a and the entire process can beexpressed in one custom interaction locus as shown in FIG. 24B.

With 15 values there is no longer a need for an interaction locus todiscriminate input values from result values or to associatedifferentiated variables because variables do not need to bedifferentiated in the first place. A transform table with more lettersis shown in FIG. 25a. The discussion has found its way back to the purevalue expression as shown in FIG. 25b that defined the example processat the start.

G. Fewer Available Values

If there are fewer available values, for example 3 values, the inputdistinctions can still be differentiated with 1 variable of three valuesand the result can still be expressed with 2 variables of three values.The only difference is that a single interaction locus can now onlyrecognize 2 unambiguous input names instead of 3 unambiguous input namesso some extra loci might be required in the expression. It should benoticed that three values are optimal for this process in that no valueexpression capacity is wasted. All the possible names for both input andresult are used.

When only two values are available it becomes necessary to encode theinput distinctions in 2 variables and the result distinctions in 4variables. Furthermore a locus now can recognize only 1 unambiguousinput name. The net result is that there are lots more variables andloci in the expression.

FIG. 26a shows the mapping for the example process with two value names.FIG. 26b shows the same process with more familiar symbols. FIG. 27 is atraditional, prior art, logic circuit expression of the example processthat recognizes the presented input names and asserts the appropriateresult names.

The logic circuit expression comprises an information processing unitwhich resolves names which are combinations of values, for example datavalues which represent various physical states. As shown, the logiccircuit comprises a plurality of logic gates or information processingmembers which also resolve combinations of values. The unit or logiccircuit further has a plurality of discreet conductors or informationtransmission elements which transmit combinations of values to and fromthe members or logic gates. The elements or conductors may beelectrical, optical, magnetic or other conductors as known in the art.Combinations of values transmitted by the elements or conductors arepresented to the members or gates and asserted from the the members orgates. Further, as shown, the unit maps from combinations of presentedinput combinations of values to combinations of asserted result values.The individual members resolve value combinations by asserting a resultvalue for each presented combinations of values, the asserted valuebeing dependent upon the particular combination of values presented.

H. NULL Again

Although it was previously stated that the examples would ignore theNULL value convention, the issue itself of expressing meaninglessnesswithin the expression could not be entirely ignored. In a traditionallogic circuit expression, the meaningfulness and meaninglessness of thedata values at the input and result interfaces of the expression areestablished by an external expression; usually the system clock. Allvalues expressed external to the expression are considered to beexpressing valid meaning on the clock edge.

Inside the logic circuit or information processing unit there is noexternal authority and variables have to express their ownmeaninglessness to the resolution of the expression themselves. This iswhy there was always a value internally assigned the meaning FALSE and avalue internally assigned the meaning DEFAULT. FALSE means “I am notmeaningful to this resolution”. DEFAULT means “I may or may not bemeaningful to this resolution depending on whether a nondefault value isasserted”.

Although the NULL value can be used to serve as a FALSE or DEFAULT valueit can also simply be added to the existing logic with its existingFALSE and DEFAULT values without disturbing the established logic of theexpression. For instance the NULL convention can be added to the binaryexample by assigning the following transform value sets to theinteraction loci.

Each locus now asserts a result value only when its input name is valid.An orderly wavefront of correct result values propagates through theexpression until the expression is asserting all nonNULL valid resultvalues. When all the result values are nonNULL the resolution iscomplete. There are no races and no spurious switching while theexpression settles to a valid result state. Both the standard logicexpression and the NULL convention expression will stabilize to the samevalue assertion state. They are both identical in terms of transformingdata but the expression with the NULL value convention can autonomouslyexpress its own completion.

Input and result names can express validity or invalidity providing anautonomous interaction coordination capability among expressions.External control expressions such as the system clock are no longerrequired.

I. The Pure Variable Representation

Two value binary expression is generally taken to be a minimal form ofexpression and might be expected to be the mentioned pure variable formof expression which uses minimal differentiation in terms of valuebecause there must be at least two values in any expression. This is notthe case however. Remember that at the beginning of this discussion theissue of autonomous control was raised and then ignored with the commentthat if an example expression was expressing process proper distinctionwith N values then it really required N+1 values to be an autonomousexpression. The traditional form of binary expression that expressesprocess proper distinction with two values is not an autonomous form ofexpression. It is really a three value form of autonomous expression.The missing control aspect of the expression that would be expressed bythe third NULL value is traditionally expressed by carefully coordinatedexternal expressions such as global clocks, delay lines or other timedevents.

For an autonomous pure variable expression with two values one valuemust be the NULL value which leaves only one value for process properdistinction. This means that all process proper distinction must beexpressed by variable differentiation. The only significance that avalue can represent is whether a variable's distinction is asserted ornot asserted; valid or invalid. A pure variable expression has the 2values; ASSERT and NULL.

The pure variable expression could be considered a one value expressionfrom the traditional point of view. Some readers may prefer to view itas a pure control expression. Since there is only one process properdifferentiation value the only valid input names that can be formed areall the same value so all that can be done by an interaction locus torecognize a formed name is to count asserted values. So a pure variableexpression is a discrete form of threshold logic.

The quickest way to grasp the pure variable expression is to compare itwith the familiar example of the half adder logic circuit in FIG. 12.The 4 variables A, B, C and S of the logic circuit each with two valuesexpress 4 process proper input distinctions and 4 process proper resultdistinctions.

FIG. 28a is the truth table for the halfadder process and FIG. 28billustrates a pure variable expression of the half adder process. Thenumber inside each interaction locus indicates how many ASSERTs arerequired to set the result variable to ASSERT. The value transform rulesets are as follows. A is ASSERT and N is NULL.

The four input and four result distinctions expressed with two variablesand two proper values in the logic circuit are now expressed with fourvariables and one proper value. There must be a presentation conventionthat only two of the input variables, one for A group and one for Bgroup can be asserted simultaneously. These two assertions will enablethe assertion of only one of the 2 threshold loci. The result of theasserting 2 threshold locus will enable the assertion of the appropriate1 threshold result loci to assert the correct result name.

FIG. 29 shows the baseline example process in a pure variableexpression. The values from the pure value expression are overlaid onthe pure variable expression to illustrate the correspondence betweenthe two expressions. Only one variable from A, B, C and one variablefrom X, Y, Z will be asserted simultaneously. This is simply aexpressional convention. This expression will assert only one of itsresult variables and expressions associated with its input name willalso assert only one of their result variables.

The essence of any process is the possible interaction relationshipsamong the existential distinctions. The measure of a process is theparticular configuration of possible interactions among a specificquantity of existential differentiations. These two expressions expressthe same process in very different ways but there is still a directcorrespondence between the pure value and the pure variable expression.There are 9 transform rules in the pure value expression and there are 9associated interaction loci in the pure variable expression. Bothexpressions express six input distinctions with nine interactionpossibilities producing nine possible result distinctions. The transformrules and the associated interaction loci express the same interactionrelationships among 15 distinctions. These two expression both expressthe same process.

If one names the variables of a pure variable expression and associatesthe names in their interaction relationships then one has a set ofvalues and their transform rules which is a pure value expression of theprocess. If one graphs the transform rules of a pure value expressionthen one gets a pure variable expression of the process.

J. Summary

This section has been a mini-excursion through process expression fromthe pure value form to the pure variable form with emphasis on theintermediate territory. It was discovered that the discrimination powerof an interaction locus, for example an information processing membersuch as a logic gate, could be considerably less than the expressivityof its input name and also that the input names to be resolved could befar larger than the input name of any single interaction locus.Consequently the coordinated cooperation of many interaction loci, eachperforming a small piece of resolution, was required in a dependentprogression of partial resolution results to express the resolution of alarger input name. This progression of partial results manifests itselfas a network of input name formation dependencies among interaction lociwhich determines the structure of the expression. Such expressions interms of directly associated interaction loci will be called directlyassociated processes (DAP). An example of a DAP is an informationprocessing unit such as a logic circuit.

The structure of name formation dependencies depends on how muchresolution a single interaction locus can provide which in turn dependson the expressional resources available and the constraints imposed onthe expression. The first example was limited to 4 values and to 4transform rule sets for the interaction loci. This required a largenetwork of name formation dependencies with lots of interaction loci. Asthe constraint on the number of transform rule sets was relaxed andtransform sets were allowed to be custom defined fewer interaction lociwere needed. Also as more values were allowed fewer interaction lociwere needed until with enough values no interaction locus was requiredat all. As fewer values were allowed more interaction loci wererequired.

The expression landscape is not smooth but is punctuated with peaks ofvarious optimalities. The significant expressional advantages accruewhen the expressional resources best match the process to be expressed.For the example process the resources phased up with the process at 3,6, 9 and 15 values. At 3 values all distinctions were expressed and allpossible names were used. With 4 values many possible names were notused so the expressivity of the available values was not fully used. At6 values the input names could be uniquely expressed so thediscrimination inefficiency of the interaction locus was reduced. At 9values both input and result could be uniquely expressed with singlevalue names making the input name discrimination much easier. At 15values all distinctions of the process were uniquely expressed and theexpression became a pure value expression. A different process wouldexhibit different phasing relationships.

All the example expressions used the same strategy of expression toresolve an input name or combination of values. Each possible input namewas individually recognized and that recognition directly generated theresult value(s) for that input name. A DAP is a input name resolver. Itdetermines or specifies which input name of a set of possible inputnames is present and generates the appropriate result for that inputname.

The input name for a DAP is presented by a fixed set of input variables.Therefore all the input names of a recognition set for a particular DAPmust be the same length. The set of input names is all the possiblecombinations of values assertable by the input variables. Any arbitraryset of input names or combinations of values can be mapped to anydesired set of result distinctions by a DAP. A DAP is deterministic inthat the same input name is always mapped to the same resultdistinctions.

A DAP is still, however, just associated variables asserting values thatinteract according to value transform rules. No new expressionprimitives needed to be postulated to achieve the expressionalcapabilities of the DAP.

An information processing system for use in manipulating and resolvingdata is constructed incorporating the NULL value. The system comprisesone or more information processing units resolving combinations ofvalues, such as a logic circuit. Each information processing circuit inturn comprises one or more information processing members, such as logicgates, for resolving combinations of allowed values, such members beingcommunicatively connected via information transmission elements, forexample electrical conductors, which transmit value (physical states)between members. Importantly the value combinations (information)comprise at least one data value and the NULL value. As previouslydiscussed, multiple data values may be utilized. Additionally, thesystem may utilize additional non-data values, other than the NULL valuesuch as the INTERMEDIATE value discussed in section V below.

The value combinations resolved via the information processing members,and transmitted via the information transmission elements may compriseat least one value, either data or non-data. Such value combinationsinclude 1) the set including standard binary data values and the NULLvalue, 2) the set including only one data value and the NULL value, 3)the set including one data value, the NULL value, and the INTERMEDIATEvalue.

In an information processing unit which comprises multiple informationprocessing members, the information processing unit maps fromcombinations of values presented to it to combinations of values itasserts. The information processing unit may perform a particular dataresolution by asserting a specific combination of values for eachcombination of presented values.

Each information processing member resolves value combinations byasserting a values for each combination of values presented to it, suchthat (1) for VALID combinations of presented values the asserted valueis a data value dependent upon the particular combination of presentedvalues, and (2) for INVALID combinations of resented values the assertedvalue is a NULL value. Alternatively, member resolution may beaccomplished by asserting a value such that (1) for VALID combinationsof presented values the asserted value is a data value dependent uponthe particular combination of presented values which remains asserteduntil the combination of presented values becomes all NULL, and (2) forall NULL combinations of presented values the asserted value is a NULLvalue which remains NULL until the combination of presented valuesbecomes VALID, thus achieving hysteresis. Finally, with respect tosystems utilizing intermediate values, the information processing memberalternatively resolves value combinations by asserting a value such that(1) for VALID combinations of presented values the asserted value is adata value dependent on the particular combination of presented values,(2) for combinations of presented values that include data values andnon-data values the asserted value is an intermediate value, and (3) forall NULL combinations of presented values the asserted value is a NULLvalue.

The information processing unit cycles through resolution andnon-resolution states to allow determination by other informationprocessing units of the information processing unit's (1) completion ofa data resolution and (2) readiness to perform another data resolution.A resolution state occurring when the information processing unit ispresented with a valid combination of values and is asserting a validcombination of values. A non-resolution state occurring when theinformation processing unit is presented with an all NULL combination ofvalues and is asserting an all NULL combination of values.

IV. Generally Configurable Process Expression

A generally configurable process is an already expressed process thatcan be configured to express any other arbitrary process. The discussionwill initially focus on the expression of any arbitrary DAP. Expressingmore complex processes than DAPs with a generally configurable processwill be evident to the reader experienced in the art.

General configurability requires expressional capabilities that a DAPcannot fulfill. Once a DAP is expressed it cannot reconfigure itself toexpress a different process. DAPs can resolve very large names but eachDAP can only resolve a specific set of possible names. The resolution ofa different set of possible names would require a differently structuredDAP. Conditionality can be added to the DAP to accommodate otherconfigurations but to accommodate all possible configurations leads tointractable combinational explosion. So a generally configurable processmust be expressive in ways that a DAP cannot accommodate.

The first requirement for general configurability is cyclic iteration.There cannot be an arbitrarily sufficient supply of interaction loci toaccommodate any arbitrary DAP and even if enough loci were availablethey could not accommodate all possible association relationships. So anarbitrary DAP cannot be completely configured by any already expressedprocess. Only part of an arbitrary DAP can be configured by an alreadyexpressed process. Therefore the expression and resolution of thearbitrary DAP must occur a piece at a time within the already expressedgenerally configurable process.

This is directly analogous to the situation of interaction loci withlimited input name resolution capabilities. Larger input names have tobe resolved in a dependent progression of interaction loci that resolvethe input name a piece at a time. A generally configurable process mustbe inherently limited in its immediate expressibility so it must expressan arbitrary process as a sequence of pieces of expression. It mustcycle through several configurations each of which contributes a partialresolution and the combination of which accumulates to a resolution ofthe complete arbitrary process.

A piece of expression might be larger or smaller depending on thecapabilities of the generally configurable process. The size of thepieces is not important. What matters is that each configuration cycleis a single configuration piece of the arbitrary process. The generallyconfigurable process can only do one configuration at a time. For thecurrent discussion considering the expression of any arbitrary DAP thepieces of configuration will be individual interaction loci.

The next requirement is the independent maintenance of values within thegenerally configurable process. There must be name formation dependencyrelationships among these pieces of configuration and hence associationrelationships among them. The input name for each interaction locus isformed from the results of two or more other interaction loci. Ifinteraction loci the are configured to resolve one piece of thearbitrary DAP must be reconfigured to resolve another piece of thearbitrary DAP then the loci cannot themselves maintain their resultvalues to form the input names of other pieces that might be configuredmany cycles in the future. The result values to form any particularinput name will be generated at different times and possibly by the sameinteraction locus. These result values that form input names for futureconfigurations must be maintained through arbitrary time periods in thegenerally configurable process separately from the interaction loci thatgenerated them. Name formation dependency association relationshipsamong interaction loci of the arbitrary DAP can no longer be expressedby direct connections among loci.

If there is a separate means of maintaining result values apart from theinteraction loci that are configured to resolve input names it followsthat there must be a means of associating these separately maintainedvalues to the proper interaction locus to be resolved and a means ofassociating the result of the interaction locus to the means of valuemaintenance. This general configurability of association relationshipsis the essence of the generally configurable process.

Each interaction of the arbitrary DAP must be individually configuredwithin the generally configurable process. Each interaction requires theassociation of a validly formed input name to an interaction locus andthe association of the result to the means of independently maintainingthat result within the generally configurable process. There must be ameans of specifying which maintained values are presented as the inputname to which interaction locus and of specifying how the result of theinteraction is maintained. There must also be a means of determiningwhen an input name is validly formed by maintained result values.

The essence of a DAP expression is the input name formation dependencyrelationships among result values and input name values and how theseformed names are resolved. The same process can be expressed by anymeans that specifies the input name formation dependencies and resolveswith the correct progression of input name formations and resolutions.

There are many means of expressing these relationships. A DAP expressesthe relationships by direct association between result values and theirdependent input name values of interaction loci. A pure value expressionexpresses the dependency relationships as value correspondences amongunique result values and uniquely named value transform rules. Agenerally configurable process must be able to express any configurationof name formation dependency relationships among name resolutionelements (interaction loci) and to resolve the configured process withthe correct progression of input name formations and resolutions.

The expression of the arbitrary DAP must be a specification of thesequence of configurations for the generally configurable process. Thisspecification can not be an inherent part of the generally configurableprocess so the supplied externally to the generally configurableprocess.

Referring to FIG. 30, a generally configurable process capable ofexpressing any arbitrary DAP must posses at least one of each type ofinteraction locus, a means of independently maintaining values and ameans of configuring sequences of association relationships between themaintained values and the interaction loci in relation to externallypresented specifications.

Several new conventions of expression are required to express thegenerally configurable process. There must be the means of independentlymaintaining values over indefinite periods. There must be the means ofassociating any maintained value with any interaction locus and theresult with any value maintenance means. The entire expression mustautonomously and continuously cycle through association configurationsin relation to the configuration specifications. The first newconvention to be defined will be a memory element which will provide themeans to independently maintain values in the generally configurableprocess.

A. The Memory Element

The first necessity is to establish islands of independent and stablevalue assertion within the larger expression. Process expression asdiscussed so far in the form of a DAP has no capability to stably asserta value independently of other expression elements. A DAP iscontinuously responding to its input and its asserted result cannotremain stable unless its presented input remains stable.

As long as an expression locality is completely dependent on externalinfluences it cannot be independently assertive. The expression localitymust be at least partially dependent on internal influences asserted bythe locality itself. This can be achieved by associating a resultvariable to an input variable as shown in FIG. 31a forming a continuousassociation loop around a specific locality of expression whichinteracting with itself will form a local interaction domain that cansustain an asserted result independently of expressions external to thatlocal domain of interaction that are providing the rest of the inputname. The value transform rule set for the locality can be arranged suchthat sometimes external influence is effective and sometimes it is not.It will be remembered that the DAP was defined to be strictlydirectional and to not have any circular association relationships. Thecircular association relationship reintroduces a form of expression thatwas carefully eliminated by the interaction locus. But this time thecircular association relationship is specifically structured throughdirectionalized elements.

The value transform rule sets will be presented in the table formatbecause many rule sets will have several input values and several resultvalues. In FIG. 31b three variables A, B and O can assert three valuesX, Y and N (NULL).

In this example in FIG. 31b, there are two input variables and oneresult variable. The result variable is associated with the inputvariable B while the input variable A can be associated with any othervariable in the larger expression. If A is NULL the result variable Ostably maintains an asserted X or Y value. When A becomes nonNULL the Ovalue is set to the value asserted by A. The value asserted by Opropagates to B and O's value assertion is locked by the interactionloop between B and O. When A becomes NULL the last asserted value isindependently maintained and stably asserted by this interaction loop.There is a time latency associated with the memory element. The valueasserted by A must be maintained long enough for the result to propogatethrough the B variable. This may or may not be significant depending onthe configuration of the larger expression.

The memory element expression convention provides an island ofindependent stable value assertion. As many memory elements as desiredcan be grouped together within a larger expression to provide for thestable maintenance of as many values as desired.

The responsibility of the group of memory elements is to independentlymaintain the assertion of result values until they are used to presentformed input names to an interaction locus. Every result is destined tobe part of an input name and it cannot be predetermined which memoryelements are to be associated with which interaction locus for allpossible arbitrary DAPs. So the asserted results of all memory elementsmust be associable with the input of each interaction locus. By the sametoken the asserted results of all interaction loci must be associablewith the input of all the memory elements. There must be a generalassociability of the interaction loci's asserted results to memoryelement inputs and of the memory element's asserted values to theinteraction loci's input.

B. The Selector Element

Selective configuration of association relationships between the memoryelements and the interaction loci can be accommodated by two expressionconventions. The first is a selector element which selects one of twoinput values to pass on as its result value. The second is a distributorelement which determines which of several destinations a result valuewill go to.

As shown in FIG. 32a the input name of the selector element is formed bythree variables. Two variables carry the candidate values to pass on andthe third carries the value that determines which variable's value ispassed on as the result value. The S variable carries the selectingvalue and the A and B variables carry the candidate input values to beasserted by the result variable O.

Referring to the transform table in FIG. 32b the input name to theselector element is VALID when S is nonNULL and the selected variable isnonNULL. The result variable O will assert the value asserted by theselected input variable. Otherwise the input name is not VALID and theresult variable O will assert NULL. It does not matter what value theunselected variable is asserting.

Although the transform definition treats the input as a three value nameto be resolved like any other DAP input name, the meta view of the inputname in terms of the convention being established must view the name ascomposed of three separate parts. Two parts (A and B) are values thatcontribute directly to the result. The third part (S) does notcontribute directly to the result value but specifies which value of theother two variables will form the result value. A and B are assertedfrom two different places in the expression. Since S is determiningwhich value plays through as the result value it is specifying whichplace A or B in the expression is associated with the place of theresult variable O. The value selecting A asserted on the variable S isthe name of the place of assertion of the variable A in the expression.Similarly the value selecting B asserted by S is the name of the placeof assertion of variable B in the expression. So the value asserted by Sis the name of a place in the expression. The variable S specifies theassociation of one place A or another place B with the place of theresult variable O in the context of the expression. S will be called aplace name variable. A and B will be called data name variables.

Referring to FIG. 33, composite input names with multiple parts withdifferent significances will be a common feature of conventions definedfrom now on. The input name of the selector element consists of a placename and two data names. As shown in FIG. 34, selector elements can becascaded to accommodate arbitrarily large choice sets and they can beganged to accommodate arbitrarily long data names.

In the cascaded tree of selector elements the place name that choosesamong the many data names is itself a multiple value name. It is formedof the same values and is no different from the data names in anyrespect except for its role and its place in the expression. A placename can be manipulated, stored and transformed just like any data name.

Each input data name A, B, C and D comes from a different place in theexpression and that place in the expression now has a name with which itcan be referred to. Presenting a particular combination of values forthe place name can mean “assert at place O the asserted result from theA place in the expression”. This ability to refer to places in theexpression by names expressed just like data names is a property thatemerges through the convention of the selector and distributor elementexpressions and is the seed of the possibility of symbolic processexpression.

C. The Distributor Element

While the selector element expresses fan in association relationshipsthat determine what- place a result value will come from, thedistributor element expresses fan out association relationships thatdetermine what place a result value will go to. A single result valuecan be associated to many places in the expression through manydistribute elements but only one of the distribute elements will bepresented with a fully valid input name and pass its input value on as aVALID result value.

The input name of the distributor element is formed by two variables asshown in FIG. 35a. One variable is the data name and the other is theplace name which determines whether the data name will be passed on asthe result value.

As shown in FIG. 35b the result of the distributor elements is NULL ifany input value is NULL. When a nonNULL value is presented on variable Aand a nonNULL value is presented on the variable D the value of variableA will be asserted as the result value. The input cannot be fully validunless the D variable is VALID. If the A variable is presented toseveral distribute elements but only one of the elements has a nonNULL Dvalue then only that distributor element will pass the value asserted bythe A variable on to whatever place in the expression the resultvariable O is directly associated with. The other distributor elementswill assert NULL result values to the places their result variable isassociated with.

Distributor elements can be grouped to accommodate arbitrarily largechoice sets and they can be ganged to accommodate arbitrarily long datanames as shown in FIG. 36.

The value of the A variable will be passed on only by the distributorelements that are presented with a VALID D variable value. Each placewithin the expression associated with the result of a distributorelement has a name with which it can be referenced. The place name canmean “deliver the result from place A to the D3 place in theexpression”.

A selector element and a distributor element together can associate anytwo places in an expression and can move a data value from anywhere inan expression to anywhere else in an expression. The movement of thedata value is controlled by the place names presented to the selectorand distributor elements. The general associability of memory elementswith interaction loci is now possible.

D. Memory-Interaction Locus Association

The memory elements and the interaction loci can be configured withselected association relationships through selector elements anddistribute elements. The selector and distributor elements allow eachmemory element and each interaction locus to be accessed by names.Assuming that the input name of each interaction locus is two valueslong a set of four place names will completely specify which two memoryelements supply the two values of the input name, which interactionlocus resolves the input name and in which memory element the resultvalue is stored. Each interaction of an arbitrary process can bespecified by a different set of four place names. FIG. 37 shows thestructure of an expression with configurable association relationships.

Assume for the time being that by some means the four place names arepresented for a time. Then they become NULL for a time then four newplace names are presented and this cycle just somehow continues. Assumealso that the input name to be resolved is prestored in the memory.

The two input variable place names select two asserted values from twomemory elements via two select elements. The selected values arepresented to two distribute elements. The interaction locus place namedirects the distribute elements to present the formed input name to oneof the interaction loci. The interaction locus resolves the input nameand asserts a result value. The asserted result value is selected by aselect element also by means of the interaction locus place name. Theresult place name then directs the result through distribute elements tothe input of a single memory element. This element stores the presentedvalue and one interaction cycle is completed.

The expression begins with all values NULL except the valuesindependently asserted by the memory elements. As soon as the placenames of the input variables are nonNULL the input names to the twoselect elements become VALID because the memory elements are alwaysasserting nonNULL values. As soon as the results of the select elementsare nonNULL and the interaction locus place name is nonNULL then theinput name to two of the distribute elements becomes VALID and theirresult values become VALID. This presents a VALID input name to one ofthe interaction loci which proceeds to resolve it. The result of thenamed interaction locus becomes nonNULL and the input name to the selectelement becomes VALID. The result of the select element becomes nonNULLand when the result variable place name is nonNULL the input to thedistribute element becomes VALID. The result of one of the distributeelements becomes nonNULL and a nonNULL value is presented to the inputof one memory element and the memory element stores the value.

The four place names become NULL and the NULL values propogatethroughout the expression. Every asserted value in the expressionbecomes NULL except the values being independently asserted by thememory elements. The result value recently presented to one memoryelement becomes NULL and all of the input values to the memory elementsare NULL. Then four new place names can be presented to the expressionand a new cycle of valid names begin flowing through the expression toresolve the next interaction.

How this cycling place name presentation might be expressed is the nextconvention to be defined.

E. The Boundary Element

A single directly associated NULL convention expression can only resolveone input name at a time and that input name must be stably maintainedon the input variables of the expression until it is fully resolved andthe expression is asserting a VALID result value. This result value mustbe stably asserted until it has fulfilled its duties of forming otherinput names in the larger expression. When the result value need nolonger be asserted by the expression itself the input name of theexpression can become NULL which will propagate NULL values through theexpression and eventually set the asserted result value to NULL. At thispoint a new input name can be presented to the expression forresolution.

The presentation of a sequence of input names to one expression by oneor more other expressions must be sensitive to the NULL-VALID state ofall involved expressions. This requires a new convention of expressionresiding between the expressions which can resolve questions concerningthe state of the several sequentially interacting expressions andmediate the transfer of names between the expressions.

The essential interexpression event is the formation of a VALID inputname. The essential concerns of a name resolving expression are when isa VALID input name presented and when is the resolution of a namecompleted. An expression can determine through its value transform ruleset when an input name is valid but it cannot determine when aresolution is completed because that depends on when the expressionsusing its result value as part of their input name no longer need it tobe asserted. A resolution is completed when the result value asserted bythe expression has effectively contributed to all of its associatedinput names. There must be an agent associated with each name resolvingexpression that can answer these concerns for each expression andmediate the formation and resolution of input names among the nameresolving expressions. Because the essential interexpression event isthe formation of input names, this agent is most conveniently viewed asa boundary element associated with the input of each name resolvingexpression. This boundary element will isolate and bound the expressionand mediate all of its name commerce with other expressions. Allexpressions that exchange sequences of names with each other mustinteract through the services of a boundary element.

The boundary element must collect the input name, determine when it isVALID, store the input name and stably present it to the resolvingexpression, determine when resolution is complete, present a NULL inputname to the resolving expression to reset the entire expression to NULL,recognize when the complete expression is reset and then collect a newinput name. All of this can be achieved with a bit of memory, theability to recognize completely NULL and completely nonNULL input namesand a familiar two variable handshake protocol between boundaryelements.

The memory is necessary to store the input name so that it can be stablypresented to the resolving expression independently of the expressionsthat asserted the pieces of the input name. A presenting expression needonly assert its result value until the input name is stored into theboundary element memory.

A completely VALID input name and a completely NULL input name are thetwo discrete boundaries of interexpression name transfer. A completelyVALID input name means that an input name is formed and can be resolved.A completely NULL input name must occur before another input name canbegin forming.

The two variable handshake protocol communicates between boundaryelements these two interexpression states of input names. The uniqueaspect of the NULL convention boundary element is that the data nameitself is one of the handshake variables with its logical states beingcompletely NULL and completely nonNULL. The data name itself is thehandshake communication from the presenter to the receiver. A singleacknowledge variable communicates from the receiver to the presenter.The following conversation summarizes the exchange. The presenter is inbold text and the receiver is in plain text.

I am presenting a name to you.

(valid result name presented to boundary element)

I have received your name

(assert acknowledge when complete input name is VALID)

I understand you have the name

(NULL result name presented to boundary element)

Thank you for the name

(unassert acknowledge when complete input name is NULL)

Referring to FIG. 38, a boundary element must consist of a memoryelement, a NULL-VALID detection element and a protocol resolutionelement. A boundary element operates via a cooperative interplay ofthese three expression elements.

1. The Memory Element

The memory element will accept and store any input name presented to itwhen the variable EN (Enable Name) is VALID (V) and stably assert thestored name. When EN is NULL (N) the externally presented input name isignored by the memory. The last stored name is stably asserted until DN(Disable Name) is VALID which forces the asserted name to NULL. NULL isasserted until a new input name is accepted by EN becoming VALID. Thename asserted by the memory element cycles between all NULL and allVALID.

2. The NULL-VALID Detection Element

The NULL-VALID detection element must establish the transitions betweencompletely VALID names and completely NULL names for both presentedinput names and names asserted by the memory. When the entire presentedinput name is VALID the variable PNV (Presented Name VALID) will becomeVALID. When the presented input name is completely NULL PNV will becomeNULL. PNV must not change its value when the presented input name ispart VALID and part NULL. When the asserted input name is VALID thevariable ANV (Asserted Name VALID) will become VALID. When the assertedinput name is completely NULL ANV will become NULL. ANV also must notchange its value when the asserted input name is part VALID and partNULL.

This transition between completely NULL and completely VALID input namescan be monitored with the expression element shown in FIG. 39a. Theresult variable is associated to the input to represent the currentstate of the determination.

Assuming an all NULL starting name the result variable 0 is NULL. Asshown in FIG. 39b O will not change to VALID until both I1 and I2 areVALID. Once O is VALID it will not change back to NULL until both I1 andI2 are NULL. So the result variable O will indicate when the input nameI1I2 has changed from all NULL to all VALID or from all VALID to allNULL. These expression elements can be cascaded to accommodate any sizeinput name as shown in FIG. 40.

IF the result variable O of the expression is NULL it will not switch toVALID until all 8 input variables are nonNULL. IF 1 input variable isNULL the result of its element will remain NULL and the next element inthe network will remain NULL and so forth. The result O will not changefrom NULL to VALID until all asserted values in the network are nonNULLwhich means that all of the input variables must be asserting nonNULLvalues. Similarly the result variable O will not change from VALID toNULL until all asserted values in the network are NULL which means thatall of the input values must be NULL. The NULL-VALID detect expressionelement insures that names presented to the boundary elements andasserted by the boundary elements are changing between completely VALIDand completely NULL and can be used to assert the values for thevariables PNV and ANV.

3. The Protocol Element

Referring FIG. 41, the protocol expression element controls the memoryand manages the handshake conversation between the boundary elements.This protocol conversation will be discussed in terms of the presenting,current and next boundary elements. The presenting boundary element isthe previous boundary element presenting an input name to the currentboundary element. The next boundary element is the succeeding boundaryelement that the current boundary element asserts its result name to.This discussion will ignore the fact that there is a name resolvingexpression between the boundary elements.

The protocol element's input variables are PNV, ANV, EN and NA. Theinput names that these variables form to the protocol element will becalled state names. PNV, ANV and EN are asserted internally to theboundary element. NA (Next Acknowledge) is the acknowledge variable fromthe next boundary element receiving the asserted result name of thecurrent boundary element. The boundary element's result variables areEN, DN and OA. OA (Own Acknowledge) is the acknowledge variable from thecurrent boundary element to the presenting boundary element that isasserting the presented input name to the current boundary element. OAof the current boundary element is NA for the presenting boundaryelement. The associated boundary elements with dependent resultvariables associated between each other constitute a sort of distributedstate machine. Each boundary element is cycling through a distinct statename sequence that depends on the results of other boundary elements.The state name transform rule set for the protocol expression ispresented in FIG. 42.

These state name transformation rules establish a necessary sequence ofstate names for each boundary element which is illustrated in FIG. 43.Assuming that the entire expression begins with a NULL presented inputname the boundary element is in state name 1 waiting to receive a VALIDpresented input name with a receive input name sequence. The firstpossible event is the presentation of a VALID input name to the boundaryelement which will cause PNV to become VALID. This will form state name5 which will set EN VALID and cause the memory to store the presentedinput name and form state name 13. EN VALID and the memory storing theinput name will eventually cause ANV to become VALID forming state name15. The result of state name 15 will set OA VALID. This is theacknowledge to the boundary element presenting the input name to thecurrent boundary element. OA VALID says to the presenting boundaryelement that the name has been received, that it no longer needs to bestably presented and that it can be set to NULL. Eventually thepresenting element will set its asserted name to NULL and the presentedinput name of the current element will become completely NULL at whichtime PNV will become NULL and state name 11 will be formed. The resultof state name 11 is to reset OA and EN to NULL forming state name 3 andthe receive input name sequence is completed.

The protocol expression will remain in state name 3 until theacknowledge NA from the next boundary element becomes VALID and thedeliver asserted name sequence is begun. The NA variable for the currentboundary element is the OA variable asserted by the next boundaryelement. NA VALID means that the next boundary element has received thename and the current boundary element can unassert its result name bysetting it to NULL. NA becoming VALID forms state name 4 which will setDN to VALID which will cause the memory to set its asserted name toNULL. The asserted name becoming NULL will cause ANV to become NULLwhich will form state name 2. When the NULL asserted name propagates tothe next boundary element and its PNV variable becomes NULL it will setits OA variable to NULL which is NA for the current boundary element.When NA becomes NULL state name 1 will be formed. The deliver assertedname sequence is completed and the current boundary element is ready toreceive another input name from the presenting boundary element.

There are two variables that are not directly or indirectly under thecontrol of the current boundary element. The presented input name maybecome VALID at anytime after a receive input name sequence is completedand NA may become valid at anytime after the asserted name becomes VALIDwhich is indicated by ANV becoming VALID. Both of these eventualitiesare accommodated by alternate state name sequences for both the receiveinput name sequence and the deliver asserted name sequence. During areceive input name sequence NA can become VALID anytime after theasserted name and ANV has become valid. The state name sequence 16, 12and 4 incorporates the VALID NA and allows the receive input namesequence to complete without a deliver asserted name sequence starting.The last state name 4 begins a deliver asserted name sequenceimmediately after the receive input name sequence is completed.

Any time after a receive input name sequence is completed or during adeliver asserted name sequence the presented input name can become VALIDand PNV will become VALID. A new presented input name cannot be receivedbefore the current asserted name has been delivered so the receive inputname sequence must be suppressed until the deliver asserted namesequence is completed. The state name sequence 7, 8, 6 and 5accommodates this by incorporating PNV VALID in the state names of thedeliver asserted name sequence. With PNV VALID the deliver asserted namesequence will end in state name 5 which will immediately begin a receiveinput name sequence.

Referring to FIG. 44, a presented input name will be received and storedin the memory elements and stably asserted until the next boundaryelement can receive it. Then the current boundary element is freed up toreceive another presented input name. Names flow through an expressionas packets from boundary element to boundary element. Between theboundary elements there can be any name resolving expression. The nameswill flow through the name resolving expression and be transformedbefore reaching the next boundary element.

The name transmission path from boundary element A to boundary element Bgoes through a name resolving expression that will perform sometransform on the name. A will complete a receive sequence and present aVALID asserted name. ANV will become VALID and the name will be asserteduntil the receive input name sequence is initiated by B. The assertedname will propagate through the name resolving expression. The initiallyNULL asserted result of the name resolving expression will at some timebecome all VALID and present a VALID input name to boundary element B.The presentation of a VALID name to the input initiates the receiveinput name sequence for B. When the receive input name sequence isinitiated OA of B is set VALID which is NA for A and which initiates thedeliver asserted name sequence for A. The asserted name of A is set toNULL and this NULL name propagates through the name resolving expressionsetting its result to NULL. The input name of 8 becomes NULL and PNV ofB becomes NULL. This causes B to complete its receive input namesequence by setting OA to NULL which is NA for A and completes thedeliver asserted name sequence for A. The name resolving expression hasbeen reset to NULL so the boundary element can now receive another inputname to assert to the name resolving expression for resolution. Theboundary element protocol will properly resolve with any arbitrary delayin the data name transmission path.

A name resolving expression bounded by boundary elements will cyclethrough completely NULL states and completely VALID states so that theNULL convention criteria that allows determination of the completion ofthe resolution of a name resolving expression is satisfied.

4. Boundary Element Association Structures

Boundary elements can be associated in various structures. The simpleststructure is a pipeline as shown in FIG. 45.

The pipeline is completely autonomous. A VALID input name presented tothe first boundary element in the pipeline will begin a sequence ofinteractions that will propagate that name from element to elementcompletely through the pipeline. As each element sees a valid input nameit will receive it and assert it to the next element in the pipeline.Several names can be simultaneously propagating just like any otherpipeline. The propagation rate of the pipeline is determined by thelongest name transmission delay between two boundary elements.

Boundary elements can be associated in a fan in configuration thatbuilds an input name from several asserted names as shown in FIG. 46.The 3 boundary elements are combining their asserted names to form theinput name for another boundary element. The receiving element will notrecognize a VALID input name until all 3 asserted names are valid. Itdoesn't matter when these asserted names became valid each one will bestably asserted until it acknowledged and its asserting boundary elementgoes through a deliver asserted name sequence. An acknowledge in theform of a VALID NA will not occur until the receiving boundary elementsees a completely valid input name. When a completely VALID input nameis presented the receiving element will initiate a receive input namesequence generating a VALID OA which is fanned out to be the NA for eachasserting boundary element. Each asserting boundary element initiates adeliver asserted name sequence and resets its asserted name to NULL.When the NULL name has been received from each asserting boundaryelement and the complete input name is NULL the receiving boundaryelement will reset OA to NULL completing its receive input name sequenceand completing the deliver asserted name sequence for each of theasserting boundary elements.

Referring to FIG. 47, a fan out association where 1 asserting boundaryelement delivers its asserted name to several other boundary elementsrequires an acknowledge collector to insure that all the receivingboundary elements have received the asserted name before resetting it toNULL.

The acknowledge collector is the same expression as the NULL-VALIDdetection element used in the boundary element. The input to thecollector are the OA acknowledge variables from each receiving boundaryelement. When all of the OA variables are VALID indicating that all ofthe receiving boundary elements have seen a VALID input name thecollector asserts a VALID result variable which is the NA variable forthe asserting boundary element. The asserting boundary element can thenset its asserted name to NULL. The result of the collector does notbecome NULL until all the OA variables from all the receiving boundaryelements are NULL indicating that the receive input name sequence forall the receiving boundary elements has been completed. The deliverasserted name sequence for the asserting boundary element is thencompleted.

Complex association structures of boundary elements can be formed byinterassociating these basic structures. Structures can be formed thatwill deadlock and livelock and otherwise misbehave but there is aninherent rationale for keeping the boundary element structures simple.If a boundary element association structure is complex in such a waythat many expression elements must be tied up maintaining their resultnames until they have formed a VALID input name then that expressionmight as well be expressed as a single directly associated process.

Boundary elements are best used to partition an expression into discreteindependently proceedable units that may be complex internally but thathave fairly straightforward and simple interfaces between them. Thisallows many expression elements to be simultaneously resolving namesincreasing the throughput of the expression. So the appropriate place inthe expression for a boundary element association is wherever there is afairly simple name association relationship.

An expression of associated boundary elements is like a chain reactionpoised to be triggered. A valid input name will trigger the progressionof events that is the expressed process. As the events proceed theexpression resets itself to be triggered again. The expression iscomplete in itself. No external driving influence such as a clock isneeded. There is nothing special or magic about the expression it isjust a specific associational structure of expressional conventionelements themselves structures of primitive expressional elements.Nothing new beyond variables, values, value transform rules and variableassociation rules has been postulated to achieve this autonomousbehavior.

F. A Generally Configurable Process

The boundary element convention completes the set of new conventionsneeded to express a generally configurable process. With the boundaryelement a sequence of directives, such as program instructions, each ofwhich includes several place names can be properly presented to thememory-interaction locus expression. FIG. 48 shows a generallyconfigurable process.

The generally configurable process expression is completed by adding tothe memory-interaction locus expression two boundary elements and amemory to maintain the directives that specify the progression ofassociation relationships. Each directive consists of the set of thefour place names discussed earlier. At the top of the expression is aloop of two boundary elements that forms the control aspects of theexpression. One boundary element maintains the name of the nextdirective and one boundary element maintains the current directive. Theboundary elements are associated in a loop that will remain activelycycling through consecutive directives as long as there is a valid nextdirective name.

Since the issue of I/O is being ignored by this example it will beassumed that the directive memory is properly set and that that theinput name to be resolved is already in the value maintenance memory.Activity is initiated when the next directive name boundary elementasserts an invalid next directive name. This name resolves the selectelement to pass the asserted contents of one of the directive memoryelements. This name becomes VALID to the current directive boundaryelement which receives the directive with a receive input name sequencewhich completes the deliver asserted name sequence of the next directivename boundary element. This frees up the next directive name boundaryelement to receive a new name. This new name is the next field of thedirective just received by the current directive boundary element. Thenext directive name boundary element receives this name, presents it tothe select element which resolves the next directive from the nameddirective memory element and asserts it to the input of the currentdirective boundary element.

Further activity is blocked until the current directive is resolvedthrough the memory-interaction locus expression. The next directiveboundary element and current directive boundary element are stuck in themiddle of their respective deliver asserted name and receive input nameprotocol sequences.

The current directive boundary element stably presents the place namesof the current directive to the memory-interaction locus expressionuntil the directive is fully resolved. This full resolution isdetermined by the presentation of a valid result value to a valuemaintenance memory element. The value maintenance memory elements cannow be defined to be partial boundary elements. Enough protocol isassociated with each memory element so that it can perform a receiveinput name sequence. Since the memory elements do not do deliverasserted name sequences the receive input name sequences are notdependent on the completion of a deliver asserted name sequence. Theresult is that whenever a valid value is presented to a valuemaintenance memory element that name will be received and stored and areceive input name sequence will be initiated without delay. The valuemaintenance memory elements, then, are protocol sinks. They receive andresolve protocol transactions but do not originate any protocoltransactions.

The assertion of OA VALID by any memory element through the OR elementmeans that a VALID result name has been received and the resolution ofthe asserted directive by the memory-interaction locus expression iscomplete. In other words the directive has done its job and can beremoved. But the directive cannot be removed until the next name hasalso been received. So the OA acknowledge from the name formation memoryelement is associated through a collector element with the OAacknowledge from the next directive name boundary element which meansthat the next directive name has been stored by next directive nameboundary element. These two acknowledges means that the entire contentsof the directive have been resolved and the directive can be unassertedand the next directive received by the current directive boundaryelement.

The current directive boundary element sets the current directive toNULL. This NULL name propagates through the memory-IL element and getsto the memory element which unasserts OA. Similarly the protocolsequence with the next directive name boundary element is completed andthe result of the collector becomes NULL. The deliver asserted namesequence for the current directive boundary element is completed and thenew directive that is already presented to its input can be received andpresented to the memory-interaction locus expression. As each directiveis presented and resolved the arbitrary DAP is resolved one interactionat a time and the result is left in the value maintenance memory.

Given a properly formed set of directives any desired structure ofassociation relationships among the available interaction loci can beexpressed. Once the expression is set up and the first directive name isinserted into the next directive name boundary element the expressionwill begin cycling through directives and resolve the expression quiteautonomously.

G. The Directive Expression

To keep the discussion in somewhat familiar territory, the binary logicexpression of the example DAP as previously discussed in relation toFIG. 26b and FIG. 27 will be used as the example for this discussion.

For this example the set of interaction loci are AND, OR and NOT. Thedata name to be resolved must be stored in the value maintenance memoryso the first step is to assign the input variables and each resultvariable in the expression the place name of a memory element. It willbe assumed for the example that the place names of the name formationmemory elements are the letters of the alphabet. An arbitrary assignmentof letter place names to results for the example is illustrated in FIG.49.

This completely defines the memory assignments necessary to express theexample DAP. There must be one directive for each interaction locus inthe DAP.

The general format for a directive is as follows.

next locus input 1 input 2 result directive name name name name name

The directive for the resolution of the input name in memory elements fand i by an AND gate with its result assigned to n would be;

AND f i n next

The results stored in f and i form the input data name for resolution bythe AND gate and the result of the resolution is stored in n. The namenext signifies the place name in the directive memory element whichcontains the next directive.

Since the generally configurable process can only resolve one directiveat a time the next step in mapping a DAP into directives is tosequentialize the progression of data name formations and resolutions.There are many sequences that will suffice. The criteria as with anyother form of expression is that each input data name is validly formedbefore it is resolved so any sequence that always generates the resultcomponents of an input data name before that input data name is resolvedis adequate and correct. One such sequence is illustrated in FIG. 50with a sequence thread passing through the example DAP.

There are lots of sequence threads that will not suffice. For instancethe reverse of the example thread would be completely wrong. No inputdata name would be validly formed when its resolution was directed. Thestructural essence of any process expression whether it's expressedsequentially or concurrently is the name formation dependencies amongthe name resolutions. The following list is the complete sequence ofdirectives to express the example DAP.

name name name directive formation formation formation directive memorymemory memory memory memory element DAP element element element elementname name name name name name 1 NOT C a 2 2 NOT D b 3 3 NOT A c 4 4 NOTB d 5 5 AND A d j 6 6 AND B c i 7 7 AND c d h 8 8 AND C b g 9 9 AND D af 10 10 AND a b e 11 11 AND e i k 12 12 AND e j l 13 13 AND f h m 14 14AND f i n 15 15 AND f j o 16 16 AND g h p 17 17 AND g i q 18 18 AND g jW 19 19 OR q p x 20 20 OR o n w 21 21 OR q p v 22 22 OR m l u 23 23 OR qo t 24 24 OR k m s 25 25 OR s t Z 26 26 OR u v Y 27 27 OR w x X NULL

The sequence of directives along with the generally configurable processand the input data form a complete process expression. A completelydifferent process be expressed by presenting a different sequence ofdirectives to the generally configurable process.

1. Enhancements

Now that the issues of local stable value expression and cyclic namepresentation have been resolved the example generally configurableprocess can be straightforwardly enhanced to a generally configurableprocess equivalent to the modern computer.

Straightforward extensions to the example expression can bring it morein line with the current vision of a processing architecture. The memoryand data paths can be widened to accommodate larger names and theinteraction loci replaced with a set of larger directly associatedprocesses such as arithmetic-logic operations to resolve the largernames. The name resolution elements are just pieces of preexpressedprocess that the arbitrary processes can use in their own expression. Aresolution element can be as small as an interaction locus or as largeas another cycling expression such as an array processor. All control isasynchronous and a resolution element can be bounded and configured toreturn its own handshake value to indicate completion.

A second alternative next directive name can be specified in thedirective and the actual next directive chosen conditionally on aresolution result value. Another means to the same end is to define thedirective memory place names to be consecutive numbers and arrange thedirectives such that the implicit next directive name is almost alwaysone greater than the current directive name. An implicit next directivename can be maintained in a small specialized memory element thatautomatically increments its value each cycle. Only alternativeconditional directive names need be explicitly specified in thedirective itself.

The value maintenance memory can also be named with consecutive numbersand values can be referenced relative to indirect relationships or evento arithmetic formulas so that the configuration of associationrelationships between the value maintenance memory and the nameresolution elements can be relative to previous configurations or evenconditional on the values of input data name from the actualityexpression of the arbitrary process. Small memory elements at variousplaces in the expression can maintain and manipulate place names forminga register set with indexing capability. Directives can indirectly referto actual place names in the memory by referring directly to thesesmaller memory places and arithmetic functions on their asserted values.

The directive memory and the data memory can be made the same memory. Itshould be evident to the knowledgeable reader that the only aspect ofcontemporary processing architectures that cannot be convenientlyaccommodated by the example is the imposition of global system clock.

With the addition of conditionality relative to the input data name inboth the determination of the next directive and the determination ofthe association relationships between value maintenance memory and theresolution elements the generally configurable process establishes a newrealm of expressivity that was not attainable with just a DAP or withthe nonconditional generally configurable process. This is thecapability to resolve indefinite length names.

2. The Advantage of Progressive Iteration

Sequential iteration while not expressionally primitive still fills animportant and essential place in the expression and resolution ofprocesses. There are input names that cannot be resolved any other wayexcept with conditional iteration.

The generally configurable process resolves an expressed arbitraryprocess by iterating through its expression a piece at a time. Theactual expression of the arbitrary process is literally composed duringresolution. Because the arbitrary process expression is effectivelycomposed piece by piece during resolution its composition can beconditionally redirected during resolution based on values in the inputdata name. The expression of the process can within limits adjust to theform of its input data in a way that a DAP could not possibly provide. ADAP cannot grow larger or smaller but a generally configurable processcan easily accommodate more or fewer iterations. A new dimension ofexpression of process is available with the conditional iterativeresolution provided by the generally configurable process.

In particular the generally configurable processor can resolve names ofindefinite length. If the size of the input data name cannot bepredetermined its resolution must be inherently iterative. The name mustbe resolved in an unpredeterminable number of partial resolution stages.Its resolution requires conditional iteration.

H. Two New Forms of Expression

Two dramatically different forms of process expression have emerged withthe generally configurable process. The first form of expression is onestep at a time sequential expression and resolution. This leads to thepossibility of conditional iteration which is a necessary form ofexpression for certain classes of processes. The second form ofexpression is the expression of the association structures andinteraction relationships of process as association relationships amongnames represented exactly the same way that input data is represented.

While both of these forms of expression are essential, neither isconceptually primitive. They are derived forms that emerge from asignificant body of expressional conventions some of which had to appealto circular variable association relationships. Association loops aroundinteraction loci provided the memory and NULL-VALID detectors with localstability of value assertion. Association loops between boundaryelements provided continuous autonomous cyclic activity. These two newforms of expression cannot be achieved without the circular associationrelationships introduced with the conventions of the generallyconfigurable process. In other words they can only be derived in termsof directly associated generally concurrent process expressions.

1. Strict Sequentiality

Strict sequentiality is generally considered to be a process expressionprimitive. Far from being primitive it has emerged late in the game byvirtue of a rather complex structure of concurrently resolving DAPs andexpression conventions with circular association relationships.

Why did strict sequentiality suddenly emerge from expression forms thatwere generally concurrent? It arose primarily because any DAP ofwhatever size can resolve only one input name at a time. Internally aDAP is generally concurrent but at its input interface it must bestrictly sequential. Each input name must be VALIDly presented andresolved then unpresented and a new input name VALIDly presented.

Strict sequentiality is also the easiest way to generally configure anyarbitrary process by configuring the association relationships of oneresolution step at a time. Each resolution step has all the resources ofthe generally configurable process devoted to it. Also a correctlysequenced expression eliminates the need for explicit VALIDation ofinput name formation. As soon as more than one resolution step isallowed to simultaneously proceed in the generally configurable processcomplicated issues of resource allocation and input name VALIDation mustbe considered.

Strict sequentiality is possible because certain forms of expression canbe reduced to a progression of independently proceedable nameresolutions. This reducibility stems primarily from the expressionalconvention of the interaction locus that imposed directionality ofinteraction influence on variable association expressions and the NULLvalue convention that imposed discrete resolution events. The onlyinfluence on an interaction locus is its input name, if the input nameis stable the interaction is a discrete resolution event that canproceed quite independently at its own pace. This makes the interactionlocus an independently resolvable unit of expression.

A directionalized process expression can be reduced to its independentlyresolvable units of expression and those units can be carried out one ata time sequentially. The sequential expression is an exact behavioralemulation of the directionalized expression. A directionalized processsuch as a DAP can be expressed and resolved with full concurrency asdirectly associated interaction loci or it can be expressed and resolvedone interaction at a time using memory to maintain result values andform input names. The resolution events of one expression are directlymappable to resolution events of the other expression and bothexpression deliver the identical final result.

Expressions that are not strictly directional such as nondirectionalizedvariable associations or direct circular association relationshipssimilar to those of the boundary element cannot be similarly reduced toa sequence of independent resolution steps because there are noindependent steps of resolution. If interactions are mutuallyinfluential in both directions then there are no clean boundaries ofinfluence and therefore no conveniently separable and independentlyproceedable pieces of the expression. These expressions are not andcannot be algorithmic.

There are many forms of process expression with nondirectional mutuallyinfluential association relationships that cannot be directly reduced toindependently proceedable resolution steps. These process expressionscan be approximately simulated in terms of discrete time steps withdifferential equations but they cannot be mapped directly toindependently proceedable resolution steps. They cannot be exactlyemulated by a sequential step by step expression. The fact that aprocess expression can be approximately simulated algorithmically doesnot mean that the process expression itself is algorithmic.

Particles in atoms and atoms in molecules are primitive variableassociation expressions that are continuously and mutually influential.In a Hopfield neural net, every resolution element receives its inputname from all the other resolution elements so that every resolution isdependent on every other resolution. These expressions rely on multiplesimultaneous associative resolution. No piece of the expression can beresolved independently from the other pieces of the expression. Itcannot be reduced to a sequence of partial resolutions that accumulateto a total resolution. These process expressions are not algorithmic andthe notion of the algorithm cannot directly encompass their resolutionbehavior.

The claim commonly expressed in discussions of cognition or artificialintelligence that any parallel expression can be sequentialized and thattherefore parallelism or concurrency can be conveniently ignored inattempts to understand the workings of human mentation is simplymistaken. While sequential resolution is essential to an important classof processes it is neither primitive nor is it universally sufficient asa form of process expression.

2. Name Relationship Expression

Perhaps the most dramatic new form of process expression that hasemerged with the generally configurable process is the expression ofprocess as deferred specifications in exactly the same way that theinput names are deferred specifications for DAPs. Process can beexpressed as a configuration of values in exactly the same way that theinput data name is expressed as a configuration of values.

As shown in the above table, a generally configurable process is aprocess that manages to defer almost all specification. From the pointof view of the generally configurable process its actuality expressionprovides almost all of the expression of the arbitrary process beingexpressed including its input data name. From the point of view of thearbitrary process the generally configurable process may express aslittle of the arbitrary process as a few interaction loci. The rest ofthe possibility expression of the arbitrary process can be expressed interms of relationships among names of places in the generallyconfigurable process. These place names are formed as combinations ofvalues in exactly the same way that input data is formed.

The names presented directly to the generally configurable process mustof course be in terms of the values and combinations of values that theselect and distribute elements recognize. A name, however, is just anexpression of correspondence whether it is a set of voltage values, astring of characters or a protein shape. Relationships amongcorrespondence references can mirror relationships among the actualthings they correspond to.

The place names in the generally configurable process can be assigned amore conveniently human readable form such as a string of alphanumericcharacters. These names are of course not understandable by thegenerally configurable process but there is a one to one mapping betweenthese names and the names that are understandable by the generallyconfigurable process. The same relationships among internal names thatare expressed by the directives can be expressed by syntacticrelationships among these external alphanumeric names. An arbitraryprocess can be conveniently expressed externally by humans in terms ofrelationships among alphanumeric names and then directly mapped into theinternal directives understandable by the generally configurableprocess.

The form of external expressions can be even further decoupled from theform of internal expressions. Since any memory element can be associatedwith any name resolution element there is no particular structure to thememory. Any internal memory name can be mapped to any external name aslong as the mapping is one to one. Therefore external names can bechosen quite arbitrarily. They do not have to correspond in any way tothe internal memory names. So external process expressions can expressassociation relationships among arbitrarily chosen external names whichcan be mapped into relationships among arbitrarily assigned internalmemory element names. There must still, however, be a directrelationship between the internal and external names of the resolutionelements. This is similar to the traditional form of external expressionknown as assembly language.

With conditional next directive and with relative memory name referenceit is possible to resolve the same group of directives representing apiece of expression multiple times each time with a different dataassociation configuration. This means that a group of directives inmemory representing a piece of expression can be treated exactly as thedirectly associated name resolution elements of the generallyconfigurable process are treated. That is, the same piece of processexpression in memory can be presented over and over with differentconfigurations of association relationships to its input data name andits result place in memory. Just like the interaction loci of theexample each resolution step can resolve a different name and put theresult at a different place in memory. But these resolution steps arefor arbitrarily defined pieces of process expressed as directives inmemory.

This means that processes expressed as hierarchically nested pieces ofprocess expression can be mapped directly into a generally configurableprocess. A group of directives can be mapped anywhere in memory and thedata to be associated with it can be mapped anywhere into memory and thecorrect association relationships among these pieces can be internallyexpressed by the generally configurable process.

The generally configurable process with generally associable memory,conditional directive sequencing and relative memory name generation isa universal process. Directly associated name resolving elements are noteven necessary. The transform rule set for any interaction locus can beexpressed in memory as a look up table accessed with relative memoryname formation. A value transform rule is just another form ofassociation relationship among names. Any directionalized processexpressed in any external form of expression that specifies associationrelationships among names can be mapped to an internal expression andresolved in a generally configurable process.

The possibility of expressing processes solely in terms of relationshipsamong names has emerged from the convention of the generallyconfigurable process. Processes can be expressed as associationrelationships among names in forms radically different from the form oftheir direct resolution. The expression of relationships among names maybe in the form of DNA, a character string, an audible utterance, apattern of neurons or a mathematical formula.

I. Summary

Several new expression conventions were introduced which combined toform the convention of the generally configurable process. The generallyconfigurable process led to the possibility of expressing processesentirely in terms of relationships among names and sponsored the firstappearance of strict sequentiality in process expression. This was allachieved with the introduction of expression conventions with closedassociation loops of directionalized expression elements which providedthe local value assertion stability and cyclic behavior required for agenerally configurable process.

No new primitives were postulated. It is all still just associatedvariables asserting values that are changing according to valuetransform rules.

Information processing units can be constructed to perform a variety offunctions, and the NULL value convention information processing systemdiscussed earlier is generally configurable, such that it canreconfigure value presentation relationships among several informationprocessing units relative to directives.

With respect to the capabilities of information processing unit functionit was first discussed that such units could perform particular dataresolution functions. Memory information processing units can store datavalues by asserting a combination of values equal to a previouslypresented combination of values. The selector information processingunit can select data values by asserting a combination of values whichare a subset of a first combination of values relative to a secondcombination of presented values. Finally, the distributor informationprocessing unit can distribute data by asserting a combination of valuesequal to a first combination of presented values as a subset of itscombination of asserted values relative to a second combination ofpresented values.

The information processing system also comprises bounding informationprocessing units to asynchronously coordinate value presentation betweeninformation processing units. The bounding elements comprise one or moreNULL-VALID detectors between information processing units fordetermining an information processing unit's (1) completion of a dataresolution and (2) readiness to perform another data resolution. Thebounding elements also comprises an information processing unit forstoring presented value combinations and an information processing unitfor communicating with other bounding means. The NULL-VALID detector isan information processing unit which asserts a NULL value when itscombination of presented values is all NULL, and continues asserting aNULL value until its combination of presented values becomes VALID. Itthen asserts a data value and continues asserting a data value until itscombination of presented values becomes all NULL.

The communication unit comprises an information processing unit forinforming all existing preceding bounding elements that a firstNULL-VALID; detector has detected a valid combination of presentedvalues, the valid combination of presented values has been stored in thestoring unit, and that all existing preceding bounding elements can nowassert an all NULL combination of values. The communication unit furtherhas means for informing all existing preceding bounding elements thatthe first NULL-VALID detector has detected an all NULL combination ofpresented values and all existing preceding bounding elements can nowassert a valid combination of values. The communication unit further hasan information processing unit for detecting that all existingsucceeding bounding elements have detected and stored a validcombination of values resulting from the valid combination of valuesstored in the storing units and asserted by the bounding elements,whereupon an all NULL combination of values can be asserted by thebounding elements. The communication unit further has an informationprocessing unit for detecting that all existing succeeding boundingelements have detected the all NULL combination of values asserted bythe bounding elements, whereupon a valid combination of values can beasserted by the bounding elements.

The generally configurable system for manipulating and resolving datacomprises at least one information processing unit for resolvingcombinations of values, at least one information processing unit forstoring combinations of values, at least one information processing unitfor configuring value presentation relationships relative to a secondcombination of values and bounding means for asynchronously coordinatingvalue presentation.

The configuring information processing units, for example selectors anddistributors, configure value presentation relationships among theresolving information processing units and the storing informationprocessing units relative to a combination of directive values which areasserted by a first bounding means and presented as the secondcombination of values of the configuring information processing units. Aresolution configuration involves the presentation of a validcombination of directive values to the configuring informationprocessing unit resulting in the presentation of a valid combination ofvalues to the storing information processing unit. A non-resolutionconfiguration involves the presentation of an all NULL combination ofdirective values to the configuring information processing unitresulting in the presentation of an all NULL combination of values tothe storing information processing unit. Data resolution is accomplishedby a progression of alternating resolution configurations andnon-resolution configurations relative to a progression of combinationsof directive values.

The information processing system further comprises a second storinginformation processing unit which asserts a plurality of combinations ofdirective values to a configuring information processing unit such thatany combination of directive values asserted by the second storinginformation processing unit can be conditionally selected forpresentation to the first bounding means relative to a secondcombination of values presented to the configuring informationprocessing unit by at least one second bounding means. Preferably thesecond bounding means is presented a combination of values asserted by aconfiguring information processing unit relative to a combination ofvalues asserted by at least one function information processing unit.

V. Completion Integrity of the NULL Value Wavefront

The configuration previously presented for expressions that are usediteratively to resolve multiple successive input names includes theexpression itself and at least one boundary element receiving theresults of the expression as shown in FIG. 51.

The role of the boundary element is to determine when the resolution ofa combination of input DATA values (an input name) being performed bythe expression is complete and also when the expression is completelyreset to NULL and ready to accept another input combination of DATAvalues to resolve. The boundary element accomplishes this by monitoringthe result values of the expression with a NULL-VALID detectorexpression. When some or all of the result values of a resolvingexpression change from all NULL values to DATA values the assertedresult of the expression is VALID and the resolution of the presentedinput DATA values is complete. It has been assumed that when the resultvalues change from all DATA to all NULL then the expression is reset andready to accept another combination of input DATA values to resolve butthis is not necessarily always the case.

For an expression that is reused iteratively to resolve successive inputnames the resolution iterations must be discretely bounded. The DATAvalues from one resolution must not get mixed up with the DATA valuesfrom a previous resolution. This is accomplished by alternatelypresenting DATA values and NULL values to the expression. Thesealternate presentations create alternating wavefronts of DATA values andNULL values through the expression.

These alternating wavefronts through the expression result in successivecycles of the expression asserting all NULL values and asserting allDATA values. There must not be any residual DATA values lingering in theexpression from a previous resolution when a new combination of DATAvalues is presented for resolution. It is the responsibility of the NULLwavefront to completely isolate successive DATA wavefronts. The entireexpression must be reliably asserting NULL values at the time ofpresentation of a new combination of input DATA values.

The expressions discussed so far can guarantee the completion integrityof a wavefront of DATA values propagating through a NULL valuedexpression. It has been implicitly assumed so far that NULL valuespresented to the input of the expression propagate through the DATAvalues of the expression just like the DATA values propagated throughthe NULL values and that when all the result values are NULL then theentire expression is NULL. But this is not always true as can be seen inthe example in FIG. 52a and FIG. 52b.

The fundamental problem is a failure of a completeness of input criteriato each interaction locus when NULL values are presented. The valuetransform set associated with each interaction locus guarantees that ifthe input and result values are all NULL then the result value will notbecome DATA until all the presented input values are DATA. This insuresthat when the result values for the interaction locus become DATA thatthe presented input values of the interaction locus are DATA. As can beseen in the transform table of FIG. 52b the interaction locus does not,however, guarantee that if the input values and the result value are allDATA that the result value will not become NULL until all of the inputvalues are NULL. So the interaction locus does not enforce acompleteness of NULL input before asserting a NULL result value.

FIG. 53 illustrates the difficulty for an expression composed ofmultiple interaction loci. Assume that all input values are DATA and theresult values are DATA. If only one input variable changes to NULL andthe others remain DATA both result variables will switch to NULL. Ifinput variable D goes NULL the result variables of interaction locus 2are set to NULL. They present NULL input values to interaction loci 3and 4 and the values of result variables S and T of 3 and 4 are set toNULL. The result values indicate that the NULL wavefront propagation iscompleted but the entire expression has not been set to NULL. The inputvariables A, B and C are still presenting DATA values to the expressionand the interaction locus 1 is still asserting a DATA value when theresult values all become NULL.

The NULL wavefront racing ahead of the DATA values can occur even if allthe input variables are set to NULL. Assume that the variable connectingloci 1 and 3 is very slow at propagating its value. The NULL value fromlocus 2 will determine the result value of result variable S. The resultvalues will indicate that the NULL value wavefront is completelypropagated and that new DATA values can be presented to the expressionbut there is still a DATA value from the old data wavefront lingering inthe expression on the 1-3 variable. This lingering DATA value cannotcause further switching of the result values associated with the NULLvalue wavefront but if it lingers long enough it could get confused withthe values of the next DATA value wavefront.

When asserting a DATA result value an interaction locus must establishthe completeness of NULL value input before asserting a NULL resultvalue. Similarly, when asserting a NULL result value an interactionlocus must establish the completeness of DATA value input beforeasserting a DATA result value. The completion integrity of the NULLvalue wavefront through DATA values of an expression must be establishedin the same sense that the completion integrity of the DATA valuewavefront through NULL values of an expression is established. Thisextra complexity requires extra specification. The extra specificationcan be in terms of more variables or in terms of more values. Bothapproaches involve different value transform sets for the interactionloci.

A. The Extra Variable Solution

The extra quantity of specification necessary to establish the integrityof the NULL value wavefront can be accommodated by keeping the samequantity of value differentiation but adding more variables to increasevariable differentiation and association. The extra variable in thiscase provides feedback association of the result value to the input ofan interaction locus as shown in FIG. 54a.

The extra variable solution makes every interaction locus a statemachine that can take account of its own result value in its behavior.The extra variable feeds the result value back to the input of theinteraction locus. What was an interaction locus with 2 input variablesnow becomes an interaction locus with 3 input variables. The valuetransform rule set has to accommodate this increase in domain. In thecase of 3 value representation with 2 DATA values and 1 NULL value theset of transform rules shown in FIG. 54b provides a generic rule setthat will accommodate any 2 DATA value transform function.

The transform rules establish that if the result value is NULL it willnot change value until both input values are DATA. If the result valueis a DATA value it will not change value until both input values areNULL. The DATA transform function is defined within the bold cells ofthe above diagram. Any desired mapping can be specified in these cells.

There is still a time issue with this configuration because once theresult value is asserted the result value must propagate to the input toestablish the stable state of the interaction locus. The result value ispropagating through other interaction loci of the expression before theinteraction locus itself is stabilized. If this back propagation time iscomparable to the time between wavefronts then the next wavefront couldarrive before the interaction locus has stabilized in relation to theprevious wavefront. To avoid this eventuality the back propagation timemust be much shorter than the time between wavefronts.

1. 2 Value Expressions with Feedback

For 2 value representation using 1 DATA value and 1 NULL value thesolution is simple and direct. An interaction locus for single DATAvalue expressions can only discriminate how many data values are presentso it is a discrete threshold. All that is necessary to create theproper state machine with 2 value locus is to feed the result value backone less times than the threshold of the locus. FIG. 55 shows a statemachine locus with a threshold of 3.

Beginning in the NULL state the locus will not assert a DATA value untilthere are 3 DATA values on its input variables. The result value is setto DATA and provides 2 more DATA values at the input of the locus. Theresult value will not become NULL until the number of asserted DATAvalues at its input falls below 3. With the 2 feedback DATA values thenumber of presented DATA values will not fall below 3 until the lastactual input variable becomes NULL. In an initial NULL state theinteraction locus does not assert a DATA result until 3 actual inputvariables present DATA values. Once the result variable asserts a DATAvalue it will not assert a NULL value until all of the actual inputvariables are asserting NULL values.

The state machine approach of feeding back the result value to an inputvariable provides the NULL-VALID detector of the boundary element withthe expression behavior it expects. When all of the result variableshaving been NULL become DATA then a resolution is completed. When all ofthe result variables having been DATA become NULL then the expression iscompletely reset and ready to receive another input name to resolve.

2. Breakdown of Completeness of Input Criteria with 1 DATA Value

With the single DATA value expression the completion criteria forpresented input and asserted results breaks down. For a thresholdinteraction locus there need be only enough DATA values presented at theinput to equal the threshold for the input to be VALID. Some of thepresented input values may remain NULL and the combination of presentedinput values will still be VALID. Similarly the asserted result valuesof an expression may not be all DATA when the asserted result isactually VALID.

Because of this inherent inability to enforce completeness of presentedinput or of asserted results the integrity of the DATA wavefront cannotbe enforced. If the VALID detector itself is a threshold function thenwhen the detector declares the results to be VALID there might still beunpropagated DATA values lingering in the expression. This can affectthe NULL value wavefront also. If a locus is very slow to assert itsDATA value and is still asserting NULL when the NULL wavefront passesthen the result values will become NULL but the DATA value from theprevious resolution will still be lingering in the expression. So thecompletion integrity of the NULL wavefront cannot be enforced either.

The difficulty has to do with the nature of the exclusively single DATAvalue expression and its limitation to threshold functions. If a singleDATA value expression pretends to be a multiple DATA value expressionwith some conventions of DATA presentation and representation theenforcement of completion integrity can be regained.

Referring to FIG. 56 composite variables with 2 encoded DATA values and1 encoded NULL value are expressed in terms of 2 real variables each ofwhich can assert 1 DATA value and 1 NULL value. Composite variable A canassert values x, y and NULL The composite variable is composed of tworeal 1 DATA value 1 NULL value variables. One variable represents x andthe other represents y. The convention is followed that only one of thereal variables can assert a DATA value at a time. If the x variableasserts a DATA value and the y variable asserts a NULL value then theasserted value for the composite variable is x. If the y variableasserts a DATA value and the x variable asserts a NULL value then theasserted value for the composite variables is y.

If only 1 variable of composite variable A asserts a DATA value and onlyone real variable of composite variable B asserts a DATA value then onlyone of the threshold 2 interaction loci will be presented with 2 DATAvalues and hence assert a DATA value itself. Only one DATA valueassertion from the threshold 2 loci will be presented to the threshold 1loci and only one of the threshold 1 loci will itself assert a DATAvalue. Thus only one real result variable will assert a DATA value forthe composite result variable O. If the convention of asserting a DATAvalue for only one real variable associated with a composite variable ata time is followed then the expression will maintain the convention onits result variables.

Completeness of input presentation can be established because exactlyone real variable from each composite variable must assert a DATA valuefor the presented input to be VALID. The completion of the resolutioncan be established because exactly one real variable of the compositeresult variable must assert a DATA value.

Composite variables with more values can be represented by simplygrouping more real variables together in a mutually exclusive DATA valueassertion convention. A composite variable asserting 3 encoded valueswould simply consist of 3 real variable asserting 1 DATA value and 1NULL value. The dual rail encoding associated with Muller speedindependent circuits is one example of 2 encoded DATA values in terms of2 variables with 1 real value and 1 NULL value. The following tableshows the correspondence between dual rail encoding and 2 DATA valuesencoded with 2 variables each asserting 1 DATA value and 1 NULL value.

 x   y x −> DATA NULL true −> 0 1 y −> NULL DATA false −> 1 0 NULL −>NULL NULL spacer −> 0 0

B. The Extra Value Solution

The extra quantity of specification necessary to establish the integrityof the NULL value wavefront can also be accommodated by keeping the samequantity of and association relationships among variables but addingmore values to achieve added differentiation. In this case an extravalue is added to represent intermediate states between DATA and NULL.

The intermediate value provides a solution with no time dependencieswhatever but the NULL-VALID detect expression must be modified toaccommodate the added value. The extra value represents intermediatestates between all DATA and all NULL. Starting with a 3 valuerepresentation with 2 DATA values and one NULL value an extra valuedesignated I (INTERMEDIATE) is added to make a total of 4 values. Aninteraction locus still has 2 input variables and 1 result variable butthe value transform rules must be modified to accommodate the extravalue in such a way that a NULL result value is asserted only when allinput values are NULL and a data result value is asserted only when allinput values are DATA. For all cases where a mix of NULL and DATA valuesare presented to the input variables the result variable must assert anINTERMEDIATE value. The following set of value transform rulesestablishes the indicated interaction locus behavior.

Any desired DATA transform can be specified by the values in the boldportion of the transform table. It can be seen directly that a DATAresult value is asserted only if both input values are DATA and that aNULL result value is asserted only if both input values are NULL. Forall other cases an INTERMEDIATE result value is asserted. The criterionfor input completion is enforced for both DATA and NULL input for eachinteraction locus.

It now becomes the responsibility of the NULL-VALID detect expression ofthe boundary element to accommodate the new INTERMEDIATE result value inits value transform rule set.

The values of the result variables begin all NULL and some may becomeINTERMEDIATE before all become DATA. Similarly beginning as all DATAsome result variables may become INTERMEDIATE before all become NULL.The NULL-VALID detect expression must still determine when all resultvalues are NULL and when all result values are DATA but it must alsoaccommodate the combinations of result values with INTERMEDIATE values.

To accommodate the INTERMEDIATE value from the interaction loci of theexpression asserting the result values the first rank of interactionloci of the NULL-VALID detect tree shown in FIG. 40 must include theINTERMEDIATE value (I) in its input value set. The set of valuetransform rules shown in FIG. 57 will accommodate result values directlyfrom the resolving expression.

Once the first rank of interaction loci have accommodated theINTERMEDIATE values then the interaction loci in later ranks need notaccommodate INTERMEDIATE values. Since the first rank interaction locido not assert INTERMEDIATE result values the interaction loci insucceeding ranks can be the NULL-VALID interaction loci previouslypresented.

1. 3 Value Expressions with INTERMEDIATE Value

In a 3 value expression with INTERMEDIATE values there is 1 DATA value,1 INTERMEDIATE value and 1 NULL value. Like the 1 DATA value interactionloci previously presented the interaction loci can recognize onlyquantities of DATA values and are strictly discrete threshold functions.The transform criteria is still the same. An interaction locus shouldassert a DATA result value only when complete DATA input is presentwhich in this case means enough presented DATA values to match or exceedthe threshold and should only assert a NULL result value when all of thepresented input values are NULL. For all other cases it should assertthe INTERMEDIATE value. The value transform rule set shown in FIG. 58implements a threshold 3 interaction locus with 3 input variables. Y isDATA, I is INTERMEDIATE and N is NULL. The value transform rule setshown in FIG. 59 implements a threshold 2 interaction locus with 3 inputvariables.

The result variables of an expression composed of such interaction locican be handled exactly as in the previous example. The first rank ofNULL-VALID interaction loci in the NULL-VALID detector expression mustaccommodate the INTERMEDIATE value from the resolving expression inexactly the same way as before.

2. 3 Value INTERMEDIATE Value Expressions Encoded with 2 Real Values

A composite variable asserting 3 encoded values; DATA, INTERMEDIATE andNULL can be represented in terms of 2 real values and 2 real variablessimilarly to the way that Muller dual rail logic encodes 3 values (2DATA values and a spacer (NULL) value on 2 real variables with 2 realvalues. The specific encoding and the logical consequences of theencoding are, however, quite different from dual rail logic. Eachencoded value is expressed in terms of 2 real variables and 2 realvalues with the following encoding.

00→NULL

01→INTERMEDIATE

10→INTERMEDIATE

11→DATA

Referring to FIG. 60 a composite threshold 2 function can be expressedin the following manner with a real threshold 1 interaction locus and areal threshold 4 interaction locus. As long as the presented real inputvalues are all 0 (NULL) the real result variables will assert 0,0 whichis a NULL encoded result value. As encoded DATA values are presented tothe composite input variables the real input variables become 1. As soona one real input variable becomes 1 the top interaction locus asserts a1 real result value. The bottom interaction locus does not assert a 1real result value until all of the real input values are 1. So while theencoded input values are changing from NULL to DATA the real inputvalues change from 0 to 1 the composite result variable asserts anintermediate encoded value until both composite variables present avalid encoded DATA input at which time the real result values become 1,1asserting an encoded result DATA value on the composite result variable.

Referring to FIG. 61a composite threshold 2 locus with 3 composite inputvariables. Several real interaction loci are required but their behavioris the same as a single interaction locus. If any two or more compositevariables A or B or C become DATA then the result composite variablewill become DATA. Otherwise the composite result variable will remainINTERMEDIATE or NULL. Once the composite result variable becomes DATA itwill remain DATA or INTERMEDIATE until all of the composite inputvariables become NULL. As can be seen from this example compositethreshold functions can be expressed for any threshold and any number ofcomposite input variables

FIG. 62 shows a threshold 3 locus with 3 composite input variables.

C. Summary

As has been shown it is essential to the integrity of the iterativeresolution process to be able to determine the completeness of theresolution of the value wavefronts. Both of the general solutionspresented above allow for the determination of the completion of boththe DATA value wavefront and the NULL value wavefront. In so doing theyinsure the integrity of the iterative resolution of succeedingcombinations of DATA values.

It should be noted that both of these solutions are completely logicalin nature. In both solutions the validity of a DATA value isdeterminable by the arrival of the DATA value itself. Neither solutionrequires any external control or timing variables and are independent oftheir own internal prorogation delays.

As many changes are possible to the embodiments of this inventionutilizing the teachings thereof, the descriptions above, and theaccompanying drawings should be interpreted in the illustrative and notthe limited sense.

That which is claimed:
 1. A switching element implementing aswitching-logic relationship between at least two signals comprising: a.an input receiving at least two input signals, where each input signalmay, at various times, assume any ore of a plurality of values, at leastone of said values being a DATA value having a switchinglogic-relationship meaning, another of said values being a NULL valuehaving no switching-logic-relationship meaning, and another of saidvalues being an INTERMEDIATE value that is distinct from the DATA andNULL values and has no switching-logic-relationship meaning; b. anoutput generating an output signal, where the output signal may, atvarious times, assume any one of a plurality of values, at least one ofsaid values being a DATA value having a switching-logic-relationshipmeaning, and another of said values being a NULL value having noswitching-logic-relationship meaning; c. a switching locus responsive tothe input signals and generating the output signal according totransform rules; d. wherein the transform rules prescribe that theoutput signal switch to a DATA value in accordance with aswitching-logic relationship of input DATA values when all input signalshave assumed a DATA value; e. wherein the transform rules prescribe thatthe output signal switch to the NULL value when all input signals haveassumed the NULL value; and f. wherein the transform rules prescribethat the output signal not switch to either the DATA value or to theNULL value when any of the input signals has assumed the INTERMEDIATEvalue.
 2. The element of claim 1 wherein an input signal may, at varioustimes, assume any of a plurality of DATA values having aswitching-logic-relationship meaning.
 3. The switching element of claim2 wherein: a. the output signal may assume an INTERMEDIATE value that isdistinct from the DATA and NULL values and has noswitching-logic-relationship meaning; and b. the transform rulesprescribe that the output signal switch to the INTERMEDIATE value whenany of the input signals has assumed the INTERMEDIATE value.
 4. Theswitching element of claim 1 wherein: a. the output signal may assume anINTERMEDIATE value that is distinct from the DATA and NULL values andhas no switching-logic-relationship meaning; and b. the transform rulesprescribe that the output signal switch to the INTERMEDIATE value whenany of the input signals is in the INTERMEDIATE value.
 5. The switchingelement of claim 1 wherein DATA, INTERMEDIATE, and NULL values of aninput signal are encoded onto a number of signal lines that is less thanthe number of values.
 6. A switching element implementing aswitching-logic relationship between at least two signals comprising: a.an input receiving at least two input signals, where each input signalmay, at various times, assume any one of a plurality of signal values,at least one of said values being a DATA value having aswitching-logic-relationship meaning, another of said values being aNULL value having no switching-logic-relationship meaning, and anotherof said values being an INTERMEDIATE value that is distinct from theDATA and NULL values and has no switching-logic-relationship meaning; b.an output generating an output signal, where the output signal may, atvarious times, assume any one of a plurality of values, at least one ofsaid values being a DATA value having a switching-logic-relationshipmeaning, and another of said values being a NULL value having noswitching-logic-relationship meaning; c. a switching locus responsive tothe input signals and generating the output signal according totransform rules; d. wherein the transform rules prescribe that theoutput signal switch to a DATA value in accordance with aswitching-logic relationship of input DATA values when a logicallycomplete number of input signals has assumed DATA values; e. wherein thetransform rules prescribe that the output signal switch to the NULLvalue when all input signals have assumed the NULL value; and f. whereinthe transform rules prescribe that the output signal does not switch toeither the DATA value or to the NULL value when (i) at least one inputsignal has assumed a non-NULL value and (ii) fewer than a logicallycomplete number of input signals has assumed DATA values.
 7. Theswitching element of claim 6 wherein: a. the number of input signals isgreater than two; and b. the switching-logic relationship of input DATAvalues is a threshold relationship.
 8. The switching element of claim 7wherein the threshold is greater than one.
 9. The switching element ofclaim 8 wherein the threshold is less than the number of inputs.
 10. Theswitching element of claim 8 wherein: a. the number of input signalsequals three; and b. the threshold equals two.
 11. The switching elementof claim 6 wherein: a. the output signal may assume an-INTERMEDIATEvalue that is distinct from the DATA and NULL values and has noswitching-logic-relationship meaning; and b. the transform rulesprescribe that the output signal switch to the INTERMEDIATE value when(i) at least one input signal has assumed a non-NULL value and (ii)fewer than a logically complete number of input signals has assumed DATAvalues.
 12. The element of claim 6 wherein an input signal may assumeany of a plurality of DATA values having a switching-logic-relationshipmeaning.
 13. The switching element of claim 6 wherein DATA,INTERMEDIATE, and NULL values of an input signal are encoded onto anumber of signal lines that is less than the number of values.
 14. Theswitching element of claim 6 wherein the switching-logic relationship ofinput DATA values is a threshold relationship.
 15. A switching elementimplementing a switching-logic relationship between at least two signalscomprising: a. an input receiving at least two input signals, where (i)each input signal may, at various times, assume any one of a pluralityof values, at least one of said values being a DATA value having aswitching-logic-relationship meaning, another of said values being aNULL value having no switching-logic-relationship meaning, and anotherof said values being an INTERMEDIATE value that is distinct from theDATA and NULL values and has no switching-logic-relationship meaning,and (ii) said DATA, INTERMEDIATE and NULL values are encoded onto anumber of signal lines that is less than the number of values; b. anoutput generating an output signal, where the output signal may, atvarious times, assume any one of a plurality of signal values, at leastone of said values being a DATA value having aswitching-logic-relationship meaning, and another of said values being aNULL value having no switching-logic-relationship meaning; c. aswitching locus responsive to the input signals and generating an outputsignal according to transform rules; d. wherein the transform rulesprescribe that the output signal switch to a DATA value in accordancewith a switching-logic relationship of input DATA values; e. wherein thetransform rules prescribe that the output signal switch to the NULLvalue when all input signals have assumed the NULL value; and f. whereinthe transform rules prescribe that the output signal does not switch toeither the DATA value or to the NULL value when (i) at least one inputsignal has assumed a non-NULL value and (ii) fewer than a logicallycomplete number of input signals has assumed DATA values.
 16. Theswitching element of claim 15 wherein: a. the number of input signals isgreater than two; and b. the switching-logic relationship of input DATAvalues is a threshold relationship.
 17. The switching element of claim16 wherein the threshold is greater than one.
 18. The switching elementof claim 17 wherein the threshold is less than the total number of inputsignals.
 19. The switching element of claim 15 wherein: a. the outputsignal may assume an INTERMEDIATE value that is distinct from the DATAand NULL values and has no switching-logic-relationship meaning; and b.the transform rules prescribe that the output signal switch to theINTERMEDIATE value when fewer than a logically complete number of inputsignals has assumed DATA values.
 20. The element of claim 15 wherein aninput signal may assume any of a plurality of DATA values having aswitching-logic-relationship meaning.
 21. The switching element of claim15 wherein: a. the number of input signals equals three; and b. thethreshold equals two.
 22. A method for processing logic signalscomprising: a. providing a plurality of input signals to a switchinglocus, where each input signal may, various times, assume any one of aplurality of values, at least one of said values being a DATA valuehaving a switching-logic-relationship meaning, another of said valuesbeing a NULL value having no switching-logic-relationship meaning, andanother of said values being an INTERMEDIATE value that is distinct fromthe DATA and NULL values and has no switching-logic-relationshipmeaning; b. generating an output signal at the switching locus, wherethe output signal may, at various times, assume any one of a pluralityof values, at least one of said values being a DATA value having aswitching-logic-relationship meaning, and another of said values being aNULL value having no switching-logic-relationship meaning; c. switchingthe output signal in accordance with input signals according totransform rules; d. wherein the transform rules prescribe that theoutput signal switch to a DATA value in accordance with aswitching-logic relationship of input DATA values when all input signalshave assumed a DATA value; e. wherein the transform rules prescribe thatthe output signal switch to the NULL value when all input signals haveassumed the NULL value; and f. wherein the transform rules prescribethat the output signal not switch to either the DATA value or to theNULL value when any of the input signals has assumed the INTERMEDIATEvalue.
 23. The method of claim 22 wherein: a. the output signal mayassume an INTERMEDIATE value that is distinct from the DATA and NULLvalues and has no switching-logic-relationship meaning; and b. thetransform rules prescribe that the output signal switch to theINTERMEDIATE value when any of the input signals is in the INTERMEDIATEvalue.
 24. The method of claim 22 wherein an input signal may, atvarious times, assume any of a plurality of DATA values having aswitching-logic-relationship meaning.
 25. The method of claim 22wherein: a. the output signal may assume an INTERMEDIATE value that isdistinct from the DATA and NULL values and has noswitching-logic-relationship meaning; and b. the transform rulesprescribe that the output signal switch to the INTERMEDIATE value whenany of the input signals is in the INTERMEDIATE value.
 26. The method ofclaim 22 wherein DATA, INTERMEDIATE, and NULL values of an input signalare encoded onto a number of signal lines that is less than the numberof values.
 27. A method for processing logic signals comprising: a.providing a plurality of input signals to a switching locus, where eachinput signal may, at various times, assume one of a plurality of signalvalues, at least one of said values being a DATA value having aswitching-logic-relationship meaning, another of said values being aNULL value having no switching-logic-relationship meaning, and anotherof said values being an INTERMEDIATE value that is distinct from theDATA and NULL values and has no switching-logic-relationship meaning; b.generating an output signal at the switching locus, where the outputsignal may, at various times, assume any one of a plurality of values,at least one of said values being a DATA value having aswitching-logic-relationship meaning, and another of said values being aNULL value having no switching-logic-relationship meaning; c. switchingthe output signal in accordance with input signals according totransform rules; d. wherein the transform rules prescribe that theoutput signal switch to a DATA value in accordance with aswitching-logic relationship of input DATA values when a logicallycomplete number of input signals has assumed DATA values; e. wherein thetransform rules prescribe that the output signal switch to the NULLvalue when all input signals have assume the NULL value; and f. whereinthe transform rules prescribe that the output signal does not switch toeither the DATA value or to the NULL value when (i) at least one inputsignal has assumed a non-NULL value and (ii) fewer than a logicallycomplete number of input signals has assumed DATA values.
 28. The methodof claim 27 wherein: a. the number of input signals is greater than two;and b. the switching-logic relationship of input DATA values is athreshold relationship.
 29. The method of claim 28 wherein the thresholdis greater than one.
 30. The method of claim 29 wherein the threshold isless than the number of inputs.
 31. The method of claim 29 wherein: a.the number of input signals equals three; and b. the threshold equalstwo.
 32. The method of claim 27 wherein: a. the output signal may assumean INTERMEDIATE value that is distinct from the DATA and NULL values andhas no switching-logic-relationship meaning; and b. the transform rulesprescribe that the output signal switch to the INTERMEDIATE value when(i) at least one input signal has assumed a non-NULL value and (ii)fewer than a logically complete number of input signals has assumed DATAvalues.
 33. The element of claim 27 wherein an input signal may assumeany of a plurality of DATA values having a switching-logic-relationshipmeaning.
 34. The method of claim 27 wherein DATA, INTERMEDIATE, and NULLvalues of an input signal are encoded onto a number of signal lines thatis less than the number of values.
 35. The method of claim 27 whereinthe switching-logic relationship of input DATA values is a thresholdrelationship.
 36. A method for processing logic signals comprising: a.providing a plurality of input signals to a switching locus, where (i)each input signal may, at various times, assume any one of a pluralityof values, at least one of said values being a DATA value having aswitching-logic-relationship meaning, another of said values being aNULL value having no switching-logic-relationship meaning, and anotherof said values being an INTERMEDIATE value that is distinct from theDATA and NULL values and has no switching-logic-relationship meaning,and (ii) said DATA, INTERMEDIATE and NULL values are encoded onto anumber of signal lines that is less than the number of values; b.generating an output signal at the switching location, where the outputsignal may, at various times, assume any one of a plurality of values,at least one of said values being a DATA value having aswitching-logic-relationship meaning, and another of said values being aNULL value having no switching-logic-relationship meaning; c. switchingthe output signal in accordance with input signals according totransform rules; d. wherein the transform rules prescribe that theoutput signal switch to a DATA value in accordance with aswitching-logic relationship of input DATA values when a logicallycomplete number of input signals has assumed DATA values; e. wherein thetransform rules prescribe that the output signal switch to the NULLvalue when all input signals have assumed the NULL value; and f. whereinthe transform rules prescribe that the output signal does not switch toeither the DATA value or to the NULL value when (i) at least one inputsignal has assumed a non-NULL value and (ii) fewer than a logicallycomplete number of input signals has assumed DATA values.
 37. Theswitching element of claim 36 wherein: a. the number of input signals isgreater than two; and b. the switching-logic relationship of input DATAvalues is a threshold relationship.
 38. The switching element of claim37 wherein the threshold is greater than one.
 39. The switching elementof claim 38 wherein the threshold is less than the total number of inputsignals.
 40. The switching element of claim 37 wherein: a. the number ofinput signals equals three; and b. the threshold equals two.
 41. Theswitching element of claim 36 wherein: a. the output signal may assumean INTERMEDIATE value that is distinct from the DATA and NULL values andhas no switching-logic-relationship meaning; and b. the transform rulesprescribe that the output signal switch to the INTERMEDIATE value when(i) at least one input signal has assumed a non-NULL value and (ii)fewer than a logically complete number of input signals has assumed DATAvalues.
 42. The element of claim 36 wherein an input signal may assume aplurality of DATA values having a switching-logic-relationship meaning.